
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00001986/00001
Material Information
 Title:
 Residual local projection methods for the darcy problem
 Creator:
 Harder, Christopher E
 Publication Date:
 2010
 Language:
 English
 Physical Description:
 xxiii, 209 leaves : ; 28 cm.
Subjects
 Subjects / Keywords:
 Groundwater flow  Mathematical models ( lcsh )
Finite element method ( lcsh ) Projection ( lcsh ) Finite element method ( fast ) Groundwater flow  Mathematical models ( fast ) Projection ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Colorado Denver, 2010.
 Bibliography:
 Includes bibliographical references (leaves 206209).
 General Note:
 Department of Mathematical and Statistical Sciences
 Statement of Responsibility:
 by Christopher E. Harder.
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 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 656365944 ( OCLC )
ocn656365944

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RESIDUAL LOCAL PROJECTION METHODS FOR THE DARCY PROBLEM
by
Christopher E. Harder
B.S., Metropolitan State College of Denver, 2003
M.S., University of Colorado Denver, 2006
A thesis submitted to the
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2010
This thesis for the Doctor of Philosophy
degree by
Christopher E. Harder
has been approved
by
V/ L Sam Welch
Date
Harder, Christopher E. (Ph.D., Applied Mathematics)
Residual Local Projection Methods For the Darcy Problem
Thesis directed by Prof. Leopoldo P. Franca
ABSTRACT
When solving the Darcy problem in its mixed form, it is wellknown that the set
of pairs of spaces chosen to approximate the velocity and vector fields is restricted by
the necessary and sufficient LBB condition. In order to circumvent this restriction,
stabilized finite element methods have been developed. In the tradition of the PGEM
methods used to stabilize the Pi/P0 element for the Darcy problem, the Residual Lo
cal Projection (RELP) methods are developed as an approach to stabilization when
approximating with equalorder linear spaces using both continuous and discontin
uous pressure interpolations. A total of four related methods are presented, two of
them being of the twolevel type. The methods are analyzed and shown to be op
timally convergent. Furthermore, the methods employing a discontinuous pressure
present a massconservative velocity. Finally, numerical tests are performed to vali
date the theory and show performance on a realworld problem.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
DEDICATION
This dissertation is dedicated to members of my family, whom I have always been
able to count on for support and encouragement through my many years of study.
ACKNOWLEDGMENT
Many thanks are owed to Dr. Leopoldo P. Franca for the opportunity to travel
and meet with some of the top researchers in the field. Without question, this results
from the fact that he resides among those at the top and I am very grateful to have had
a chance to work with him.
A debt of gratitude I could never hope to repay is owed to Dr. Frederic Valentin
for his diligent and exceptionally patient oversight and contribution to the develop
ment of this work.
In the course of my studies here, I have met many wonderful people who have
enriched my life professionally and personally. In particular, thanks to Dr. Bedrich
Sousedik and Minjeong Kim for the many nights of studying and discussions (and
beer!). Thanks also to the other students, too numerous to list, with whom I have
shared many nice experiences and whom I now count among my friends. I am also
grateful to the department and its professors for the chance to gain insight and knowl
edge from the interesting classes I have attended over the years.
I would finally like to thank my friends and family for their patience during my
years of academic distraction. In particular, thanks to my mother, sister, and grand
parents for their longstanding support, my father for his curiosity, Diane for pushing
me when things were tough, and my friends in general for offering a space to be
relaxed. This would have been impossible without all of you!
CONTENTS
Figures.................................................................... x
Tables...................................................................xvii
Chapter
1. Introduction........................................................... 1
1.1 Preliminary notation ................................................ 7
1.2 The Problem.......................................................... 8
1.2.1 The Darcy Problem.................................................... 9
1.2.2 Simplifying assumptions............................................. 10
1.2.3 Weak form of Darcys problem........................................ 11
1.3 Discretization and approximation.................................... 13
1.3.1 Notation related to discretization.................................. 13
1.3.2 Numerical approximations............................................ 16
2. The RELP methods...................................................... 21
2.1 Preliminaries........................................................22
2.1.1 A few operators..................................................... 22
2.1.2 A few subspaces......................................................23
2.2 The enriching approach: Motivation...................................25
2.3 The enriching approach: Defining appropriate subspaces...............29
2.4 The enriching approach: Derivation of the methods................... 31
2.4.1 Local problems in the method........................................ 32
vi
2.4.2 Solutions to the local problems........................................35
2.4.3 Two general methods....................................................38
2.5 Existence and Uniqueness Results.......................................40
2.5.1 Some auxiliary results.................................................40
2.5.2 Existence of a unique solution for the symmetric general method .... 43
2.5.3 Existence of a unique solution for the general method..................44
2.6 Explicit methods formed from the general methods.......................46
2.6.1 Methods with analyticallydefined terms................................46
2.6.2 Methods requiring a twolevel solution ................................48
3. Error Analysis of the General Methods...................................50
3.1 Interpolation results..................................................51
3.2 Consistency results....................................................56
3.3 Mass conservation property.............................................60
3.4 Error estimates........................................................61
3.4.1 Estimates for the symmetric general method.............................62
3.4.2 Estimates for the general method.......................................79
3.4.3 Results for a general g e //'(ft)..................................... 82
3.5 Concluding remarks.....................................................85
4. Error Analysis of the Fully Discrete Methods ...........................86
4.1 The Fully Discrete Methods ........................................... 87
4.2 Numerical analysis of the fully discrete methods.......................91
4.2.1 A relationship between two velocities..................................91
4.2.2 Approximate enrichment solutions.......................................94
4.2.3 Wellposedness results.................................................98
vii
4.2.3.1 Wellposedness of the fully discrete symmetric method...............99
4.2.3.2 Wellposedness of the fully discrete full method ..................100
4.2.4 Consistency results................................................102
4.2.5 Convergence results................................................108
4.2.5.1 Convergence results for the symmetric fully discrete method........108
4.2.5.2 Convergence results for the fully discrete full method.............118
5. Numerical Experiments...................................................122
5.1 An analytical problem.................................................124
5.1.1 The symmetric methods..............................................125
5.1.1.1 Methods with an analytic Mk.......................................125
5.1.1.2 Methods with a nonanalytic Mk.....................................131
5.1.2 The full methods...................................................141
5.1.2.1 Methods with an analytic Mk........................................142
5.1.2.2 Methods with a nonanalytic Mk.....................................144
5.2 A second analytical problem.........................................147
5.2.1 The symmetric methods .............................................148
5.2.1.1 Methods with an analytic Mk........................................148
5.2.1.2 Methods with a nonanalytic Mk.....................................151
5.2.2 The full methods...................................................159
5.2.2.1 Methods with an analytic Mk........................................159
5.2.2.2 Results with a nonanalytic Mk.....................................161
5.3 The quarter 5spot problem, constant permeability.......................163
5.3.1 The symmetric methods..............................................164
5.3.1.1 Methods with an analytic Mk.......................................164
viii
5.3.1.2 Methods with a nonanalytic Mk..................................169
5.3.2 The full methods.................................................174
5.3.2.1 Methods with an analytic Mk.....................................174
5.3.2.2 Methods with a nonanalytic Mk...................................'76
5.4 The quarter 5spot problem, checkerboard permeability..............178
5.4.1 The symmetric methods............................................179
5.4.1.1 Methods with an analytic Mk......................................'80
5.4.1.2 Methods with a nonanalytic Mk...................................'82
5.4.2 The full methods.................................................183
5.4.2.1 Methods with an analytic Mk......................................'83
5.4.2.2 Methods with a nonanalytic Mk...................................'86
6. Conclusion............................................................187
Appendix
A. DEFINITIONS AND THEOREMS..............................................190
A.l Useful Definitions.................................................190
A.2 Various Useful Inequalities..........................................190
A. 3 Some notes on an enrichment velocity...............................192
B. SOME THEORY FOR OPERATOR PROBLEMS....................................196
B.l Theory for symmetric problems .......................................196
B.2 Theory for nonsymmetric problems....................................199
References...............................................................206
IX
FIGURES
Figure
1.1 A saturated porous medium with pressure held higher on the left than the
right so that flow is directed toward the right......................... 2
1.2 The normal vector....................................................... 14
1.3 The lefthand plot shows how the singular values are distributed, with
the xaxis showing the values (no meaning is assigned to the paxis).
The righthand plot shows the value of the singular values along the y
axis. In this case, the xaxis indicates the position of the singular values
in a vector ordering them from largest to smallest, with a total of 1023.. 17
1.4 A sample crisscross mesh and a curve showing better than 0(H2) con
vergence for pH in the L2norm on a family of this type of mesh.......20
5.1 The mesh used when using a refinement at the second level with linear
interpolation.............................................................123
5.2 A typical structured (left, H and unstructured (right, H mesh. 125
5.3 Convergence plots using continuous pressure interpolation for problem 1
(structured mesh).........................................................126
5.4 Contour plots of the absolute value of velocity Ui (left figure) and pres
sure pi (right figure) using continuous pressure interpolation for problem
1 (structured mesh).......................................................127
5.5 Convergence plots using for the symmetric methods with analytic Mk
and continuous pressure interpolation for problem 1 (unstructured mesh). 128
x
5.6 Convergence plots using discontinuous pressure interpolation for prob
lem 1 (structured mesh)...............................................129
5.7 Convergence plots for the symmetric method with analytic Mk and dis
continuous pressure interpolation for problem 1 (unstructured mesh). . 130
5.8 Convergence plots using continuous pressure interpolation for problem 1
(structured mesh, quadratic second level)................................134
5.9 Convergence plots using continuous pressure interpolation for problem 1
(unstructured mesh, quadratic second level)..............................135
5.10 Convergence plots using discontinuous pressure interpolation for prob
lem 1 (structured mesh, quadratic second level)..........................139
5.11 Convergence plots using discontinuous pressure interpolation for prob
lem 1 (structured mesh, quadratic second level)..........................140
5.12 Contour plots of the absolute value of velocity (left figure) and pres
sure pi (right figure) using continuous pressure interpolation for problem
2 (structured mesh)........................................................149
5.13 Contour plots of absolute value of velocity Ui and n(ui) for the method
with analytic Mk and continuous pressure interpolation (structured mesh). 165
5.14 Elevation and contour (with velocity vector field) plots of pressure pi +
for the method with analytic Mk and continuous pressure interpola
tion (structured mesh).....................................................166
5.15 Contour plots of absolute value of velocity U\ and 7r(ui) for the method
with analytic Mk and continuous pressure interpolation (unstructured
mesh).......................................................................166
xi
5.16 Elevation and contour (with velocity vector field) plots of pressure p\ +
p^1 for the method with analytic Mk and continuous pressure interpola
tion (unstructured mesh)..................................................167
5.17 Contour plots of absolute value of velocity Ui and 7r(iti) for the method
with analytic Mk and discontinuous pressure interpolation (structured
mesh).....................................................................167
5.18 Elevation and contour (with velocity vector field) plots of pressure pi +
Pe1 + Pc fr the method with analytic Mk and discontinuous pressure
interpolation (structured mesh)...........................................168
5.19 Contour plots of absolute value of velocity ui and n(ui) + uf for the
method with analytic Mk and discontinuous pressure interpolation (un
structured mesh)..........................................................168
5.20 Elevation and contour (with velocity vector field) plots of pressure pi +
Pe1 + Pe fr the method with analytic Mk and discontinuous pressure
interpolation (unstructured mesh).........................................169
5.21 Contour plots of absolute value of velocity Ui and Uh for the method
with analytic Mk and continuous pressure interpolation (structured mesh). 170
5.22 Elevation and contour (with velocity vector field) plots of pressure pn
for the method with analytic Mk and continuous pressure interpolation
(structured mesh).........................................................170
5.23 Contour plots of absolute value of velocity u\ and uh for the method
with nonanalytic Mk and continuous pressure interpolation (unstruc
tured mesh)...............................................................171
xii
5.24 Elevation and contour (with velocity vector field) plots of pressure pu for
the method with nonanalytic Mk and continuous pressure interpolation
(unstructured mesh).......................................................171
5.25 Contour plots of absolute value of velocity Ui and Uh for the method
with nonanalytic Mk and discontinuous pressure interpolation (struc
tured mesh)...............................................................172
5.26 Elevation and contour (with velocity vector field) plots of pressure pH
for the method with nonanalytic Mk and discontinuous pressure inter
polation (structured mesh)................................................172
5.27 Contour plots of absolute value of velocity u\ and uH for the method
with nonanalytic Mk and discontinuous pressure interpolation (un
structured mesh)..........................................................173
5.28 Elevation and contour (with velocity vector field) plots of pressure ph
for the method with nonanalytic Mk and discontinuous pressure inter
polation (unstructured mesh)..............................................173
5.29 Contour plots of absolute value of velocity ux and 7r(ui) + uf for the
full method (a = 0.1) with analytic Mk and discontinuous pressure
interpolation (structured mesh)...........................................174
5.30 Elevation and contour (with velocity vector field) plots of pressure pi +
Pe1 + Pe fr the methd (a = 0.1) with analytic Mk and discontin
uous pressure interpolation (structured mesh).............................175
5.31 Contour plots of absolute value of velocity U\ and 7r(ui) + uf for the
full method (a = 0.1) with analytic Mk and discontinuous pressure
interpolation (unstructured mesh).........................................175
xiii
5.32 Elevation and contour (with velocity vector field) plots of pressure p\ +
Pe1 + Pe fr the full method (a = 0.1) with analytic Mk and discontin
uous pressure interpolation (unstructured mesh)................176
5.33 Contour plots of absolute value of velocity U\ and ux + u^ + uf for the
full method (a = 0.1) with nonanalytic Mk (structured mesh, quadratic
second level)..................................................................176
5.34 Elevation and contour (with velocity vector field) plots of pressure pi +
pe +P? for the full method (a = 0.1) with nonanalytic Mk (structured
mesh, quadratic second level)..................................177
5.35 Contour plots of absolute value of velocity Uj and U\ + u^' + uf for the
method with nonanalytic Mk and discontinuous pressure interpolation
(unstructured mesh, quadratic second level)..............................177
5.36 Elevation and contour (with velocity vector field) plots of pressure pi +
Pe1 + Pe fr the method with nonanalytic Mk and discontinuous pres
sure interpolation (unstructured mesh, quadratic second level)...........178
5.37 Regions in the checkerboard problem, a = 1 in zones II and IV, and
o = 0.001 in zones I and III.............................................179
5.38 Elevation plots of pressures pi and p\ +p^ +pf for the method with ana
lytic Mk and discontinuous pressure interpolation for the checkerboard
problem (structured mesh)................................................180
5.39 Contour plot (with vector field of the corresponding velocity) of pres
sures pi and pi + pM + p^ for the method with analytic Mk and discon
tinuous pressure interpolation for the checkerboard problem (structured
mesh)....................................................................181
xiv
5.40 Elevation plots of the pointwise Euclidean norm of ui and Uy+u^ + u
for the method with analytic Mk and discontinuous pressure interpola
tion for the checkerboard problem (structured mesh)..................181
5.41 Crosssections of the pointwise Euclidean norm of U\ and U\ + u%f + u
at x2 = .25 for the symmetric method with analytic M k and discontinu
ous pressure interpolation for the checkerboard problem (structured mesh). 182
5.42 Elevation plots of pressures pi and Pi + p^ + pf for the method with
nonanalytic Mk (quadratic second level) and discontinuous pressure
interpolation for the checkerboard problem (structured mesh).182
5.43 Contour plot (with vector field of the corresponding velocity) of pres
sures pi and Pi + p + Pe for the method with nonanalytic Mk
(quadratic second level) and discontinuous pressure interpolation for the
checkerboard problem (structured mesh).................................183
5.44 Elevation plots of pressures pi and pt + p^ + pf for the method full
method (a = 0.1) with analytic Mk and discontinuous pressure inter
polation for the checkerboard problem (structured mesh)................184
5.45 Contour plot (with vector field of the corresponding velocity) of pres
sures pi and Pi + p^1 + p^ for the full method (a 0.1) with analytic
Mk and discontinuous pressure interpolation for the checkerboard prob
lem (structured mesh)..................................................184
5.46 Elevation plots of the pointwise Euclidean norm of u\ and u\ +u^ +
for the full method (a = 0.1) with analytic Mk for the checkerboard
problem (structured mesh)..............................................185
xv
5.47 Crosssections of the pointwise Euclidean norm of u\ and U\ I u^ + uf
at x2 .25 for the full method (a = 0.1) with analytic Mk and discon
tinuous pressure interpolation for the checkerboard problem (structured
mesh).....................................................................185
5.48 Elevation plots of pressures pi and p\ +p^' +pj? for the full method with
nonanalytic Mk (a = 0.1) for the checkerboard problem (structured
mesh, quadratic second level)...........................................186
5.49 Contour plot (with vector field of the corresponding velocity) of pres
sures pi and pi + p^ + Pe for the full method with nonanalytic Mk
(q = 0.1) for the checkerboard problem (structured mesh, quadratic sec
ond level)................................................................186
xvi
TABLES
Table
5.1 Table of convergence results for the symmetric method with analytic
M k and continuous pressure interpolation for problem 1 (structured mesh). 127
5.2 Table of convergence results for the symmetric method with analytic
Mk and continuous pressure interpolation for problem 1 (unstructured
mesh).................................................................128
5.3 Table of convergence results for the symmetric method with analytic
Mk and discontinuous pressure interpolation for problem 1 (structured
mesh).................................................................129
5.4 Local mass conservation for the symmetric method with analytic Mk
and discontinuous pressure interpolation for problem 1 (structured mesh). 130
5.5 Table of convergence results for the symmetric method with analytic
Mk and discontinuous pressure interpolation for problem 1 (unstruc
tured mesh)...........................................................131
5.6 Local mass conservation for the symmetric method with analytic Mk
and discontinuous pressure interpolation for problem 1 (unstructured
mesh).................................................................131
5.7 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (structured
mesh, linear second level).............................................132
xvu
5.8 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (unstructured
mesh, linear second level)...............................................133
5.9 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (structured
mesh, linear second level w/ refinement).........................133
5.10 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (unstructured
mesh, linear second level w/ refinement).........................134
5.11 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (structured
mesh, quadratic second level)..............................................135
5.12 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 1 (unstructured
mesh, quadratic second level)..............................................136
5.13 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (structured
mesh, linear second level).................................................136
5.14 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (unstruc
tured mesh, linear second level)...........................................137
5.15 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (structured
mesh, linear second level w/ refinement).................................138
xviii
5.16 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (unstruc
tured mesh, linear second level w/ refinement)................139
5.17 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (structured
mesh, quadratic second level).................................140
5.18 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 1 (unstruc
tured mesh, quadratic second level)...........................141
5.19 Table of convergence results for the full method (a = 0.1) with an ana
lytic Mk and discontinuous pressure interpolation for problem I (struc
tured mesh).............................................................142
5.20 Performance of the full method with analytic Mk and discontinuous
pressure interpolation for problem 1 with different values of a (structured
mesh)...................................................................142
5.21 Local mass conservation for the full method with an analytic Mk (<* =
0.1) and discontinuous pressure interpolation for problem 1 (structured
mesh, quadratic second level)...........................................143
5.22 Table of convergence results for the full method (a = 0.1) with an an
alytic Mk and discontinuous pressure interpolation for problem 1 (un
structured mesh)........................................................143
5.23 Performance of the full method with analytic Mk and discontinuous
pressure interpolation for problem 1 with different values of a (unstruc
tured mesh).............................................................144
xix
5.24 Local mass conservation for the full method with an analytic Mk (a =
0.1) and discontinuous pressure interpolation for problem 1 (unstructured
mesh).........................................................144
5.25 Table of convergence results for the full method (a = 0.1) with non
analytic Mk and discontinuous pressure interpolation for problem 1
(structured mesh, quadratic second level).....................145
5.26 Performance of the full method with nonanalytic Mk and discontinu
ous pressure interpolation for problem 1 with different values of a (struc
tured mesh).............................................................145
5.27 Local mass conservation for the full method with nonanalytic Mk (a =
0.1) and discontinuous pressure interpolation for problem 1 (structured
mesh, quadratic second level)...........................................146
5.28 Table of convergence results for the full method (a = 0.1) with non
analytic Mk and discontinuous pressure interpolation for problem 1 (un
structured mesh, quadratic second level)................................146
5.29 Performance of the full method with nonanalytic Mk and discontinu
ous pressure interpolation for problem 1 with different values of a (un
structured mesh)........................................................147
5.30 Local mass conservation for the full method with nonanalytic Mk (a =
0.1) and discontinuous pressure interpolation for problem 1 (unstructured
mesh, quadratic second level)...........................................147
5.31 Table of convergence results for the symmetric method with analytic
Mk and continuous pressure interpolation for problem 2 (structured mesh). 148
xx
5.32 Table of convergence results for the symmetric method with analytic
M.k and continuous pressure interpolation for problem 2 (unstructured
mesh)..........................................................149
5.33 Table of convergence results for the symmetric method with analytic
Mk and discontinuous pressure interpolation for problem 2 (structured
mesh)..........................................................150
5.34 Table of convergence results for the symmetric method with analytic
Mk and discontinuous pressure interpolation for problem 2 (unstruc
tured mesh)..............................................................151
5.35 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (structured
mesh, linear second level)...............................................151
5.36 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (unstructured
mesh, linear second level)...............................................152
5.37 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (structured
mesh, linear second level w/ refinement).................................152
5.38 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (unstructured
mesh, linear second level w/ refinement).................................153
5.39 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (structured
mesh, quadratic second level).............................................153
XXI
5.40 Table of convergence results for the symmetric method with nonanalytic
Mk and continuous pressure interpolation for problem 2 (unstructured
mesh, quadratic second level)...........................................154
5.41 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (structured
mesh, linear second level)..............................................154
5.42 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (unstruc
tured mesh, linear second level)........................................155
5.43 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (structured
mesh, linear second level w/ refinement)......................156
5.44 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (unstruc
tured mesh, linear second level w/ refinement)..........................157
5.45 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (structured
mesh, quadratic second level)...........................................158
5.46 Table of convergence results for the symmetric method with nonanalytic
Mk and discontinuous pressure interpolation for problem 2 (unstruc
tured mesh, quadratic second level).....................................159
5.47 Table of convergence results for the full method (a = 0.1) with an ana
lytic Mk and discontinuous pressure interpolation for problem 2 (struc
tured mesh)..............................................................160
xxu
5.48 Local mass conservation for the full method with an analytic Mk (a =
0.1) and discontinuous pressure interpolation for problem 2 (structured
mesh)..................................................................160
5.49 Table of convergence results for the full method (a = 0.1) with an an
alytic Mk and discontinuous pressure interpolation for problem 2 (un
structured mesh).......................................................161
5.50 Local mass conservation for the full method with an analytic Mk (<* =
0.1) and discontinuous pressure interpolation for problem 2 (unstructured
mesh)..................................................................161
5.51 Table of convergence results for the full method (a = 0.1) with non
analytic Mk and discontinuous pressure interpolation for problem 2
(structured mesh, quadratic second level)..............................162
5.52 Table of convergence results for the full method (a = 0.1) with non
analytic Mk and discontinuous pressure interpolation for problem 2 (un
structured mesh, quadratic second level)...............................163
5.53 Local mass conservation for the symmetric methods using discontinu
ous pressure interpolation for the quarter 5spot problem with constant
permeability...........................................................164
5.54 Local mass conservation for the full methods (a = 0.1) for the checker
board quarter 5spot problem.................................................174
5.55 Local mass conservation for the symmetric methods using discontinuous
pressure interpolation for the checkerboard quarter 5spot problem. ... 180
5.56 Local mass conservation for the full methods (a = 0.1) for the checker
board quarter 5spot problem.................................................183
xxiii
1. Introduction
The study of the flow of a fluid through a porous medium is of particular interest
in many fields of research. In the study of ground water contamination plumes or the
extraction of oil, the porous medium is the ground beneath our feet. This ground may
be composed of rock, sand, soil, clay, or any combination of these, and is far from
a solid, impermeable mass. Instead, gaps exists throughout, making space for oil or
aquifers, etc. Naturally, the fluid in question needs to move through this medium in
some way, leading us to question how best to model this. Furthermore, once a model
is in hand, how do we get solutions from the model which are meaningful?
Suppose we are given a porous medium which is completely saturated with a
fluid (e.g., water, oil, etc.) which is flowing through it (see Figure 1.1). One model
for describing such flows is attributed to Darcy, and in the case of a fluid in lami
nar flow, this seems to be a good model. In its primal form, this model, also known
as the Darcy problem, consists of a Poisson equation for pressure. Although this
problem may be effectively solved using standard Galerkin techniques, it is often
the flow velocity, which is related to the gradient of the pressure, is typically the
quantity of interest. For instance, a common model of the miscible transport of an
underground contaminant in water is a nonlinear system composed of the Darcy prob
lem and a diffusionconvectionreaction equation with coefficients depending on the
locally conservative flow velocity (from the Darcy portion). Furthermore, the numer
ical approach should be mass conserving in order to be physically consistent, again
requiring an accurate velocity profile. Therefore, methods which solve for the veloc
1
6
&
Figure 1.1: A saturated porous medium with pressure held higher on the left than the
right so that flow is directed toward the right.
ity directly, as opposed to obtaining it through differentiation, are of intereset. This
leads us to consider solving the Darcy problem in its mixed form.
Approximation for the mixed form is not without restrictions. When approaching
the solution numerically, one must ensure the matrix equation has a unique solution,
a property which is not directly inherited from the continuous problem. The basic
requirement is the LBB condition (or infsup condition) [14], which describes the re
lationship that must exist between the pressure and velocity approximation spaces. In
addition to issues related to invertibility, the value of using higherorder polynomials
is questionable when the exact solution doesnt have very nice regularity properties
(a typical property of all finite element methods). In fact, loworder, stable elements
which are locally mass conservative exist in a delicate balance, and many intuitive
choices simply fail to satisfy all three properties.
In order to satisfy the LBB condition, yet still use loworder approximation
spaces, one may consider using loworder RaviartThomas (RT) or BrezziDouglas
Marini (BDM) elements for discretization of the velocity. (See [34] and [15], respec
2
tively. See also [12, 22, 20] for a nice discussion, as well as [4] for an implementa
tion of loworder RT elements in Matlab.) These elements are constructed to yield
velocities with continuous normal components and (possibly) discontinuous tangen
tial components across mesh faces, and have the advantage of being locally mass
conservative and LBBstable when combined with pressure spaces of discontinuous
polynomials of order one less than the velocity space. However, the approach leads
to systems which require extra effort to solve efficiently [17].
Other loworder elements are of interest. Unfortunately, many of the most intu
itive choices simply fail to satisfy the LBBcondition. The nodally defined Pi/Po,
Pi/Pi, and Pi/Pf3C (the last element indicating linear, continuous velocity together
with linear, discontinuous pressure, respectively) are not stable pairs. However, inter
est in them persists because they are:
simple to program and they exist in standard codes;
loworder, thereby keeping the number of degrees of freedom, and therefore
the overall cost of computation, down.
Many approaches consider stabilizing such pairs (among other pairings). One
possibility is the consistently stabilized methods of [31, 29] for the Darcy problem
using any combination of standard, nodallydefined elements for the velocity and
pressure approximations. The approach is quite elegant and the stability coefficient
is simply equal to i. (See [18] fora more thorough analysis). However, the approach
does not give a locally mass conservative velocity when continuous velocities are
considered.
3
Contributions from the area of leastsquares finite element methods exist as well.
Inspired by work stabilizing the Stokes problem (see [21, 10]), elements not satisfying
the LBB condition were stabilized for the mixed form of the Darcy problem in [11]
through the use of local projections of the pressure polynomial solution of order k into
the discontinuous space of polynomials of order k 1. In fact, this type of projection
will be one of the key elements of the stabilization presented in this work.
None of the above considered the use of the simplest, although unstable, Pi/Po
element. A stabilization for this element has been considered in [5, 6, 2] in the frame
work of PetrovGalerkin Enriched methods (PGEM). PGEM has been used to stabi
lize discretizations of several operators (see, e.g., [25,7, 5]). Motivated by the notion
that instabilities in numerical solutions arise from an inability of the mesh to resolve
fine scales present in the true solution, the underlying philosophy is to decompose the
solution space into a polynomial portion and a fine (or sub) scale portion, as done
in the Variational Multiscale Method (VMS) [28] and ResidualFree Bubble Method
(RFB) (see, e.g., [19]). Through a static condensation procedure, the fine scale solu
tions are then defined in terms of local residuals of the operator. The distinguishing
characteristic for PGEM is its stabilization via the incorporation of edge residuals into
the fine scale solutions, achieved by choosing the boundary conditions for the local
problem in terms of these edge residuals. When applied to the reactiondiffusion op
erator, the imposition of such nonhomogeneous boundary conditions stands in con
trast to the RFB method, which requires the local solutions to satisfy homogeneous
boundary conditions. Although very successful, RFBs still presented nonphysical
oscillations in the solution in some numerical examples which are thought to be the
result of the nonphysical boundary condition. The PGEM for this operator may be
4
seen to be as a generalization of the RFB approach which seeks to replace the homo
geneous boundary conditions with more physical ones. The resulting methods have
shown an improvement in numerical tests (see [25, 24]).
Returning to the stable PGEM methods based upon the Pi/P0 element used in
the mixed Darcy problem, we see the methods have many appealing characteristics:
1. They are derived via an enrichment of the standard approximation spaces.
2. They use loworder, nodallydefined elements readily available in finite element
codes.
3. Symmetric methods are available in [6, 2].
4. The continuity of the velocity is relaxed to simply the continuity across edges
in the normal direction thanks to one portion of the enrichment solutions.
5. The derivation indicates a locally massconservative velocity.
6. Optimal convergence is achieved, and superconvergence is observed in some
numerical examples.
Based upon the success with such methods derived in the PGEM approach, fur
ther exploration was considered warranted, and the ideas have been extended to the
Stokes operator using, in particular, the Pi/Pi and Pi/Pflsc elements [8], The present
work considers a new extension of the PGEM to the unstable Pi/Pi and Pi/Pflsc el
ements used in the Darcy problem. The methods considered will be derived in the
PGEM framework, and are called Residual Local Projection (RELP) methods. They
may in some sense be viewed as a combination of ideas in [31, 18] (consistent sta
bilization using the residual of the momentum equation), [11] (use of the projection
5
into the piecewise constants), and importantly [5, 2, 6] (use of edge residuals to drive
enrichment solutions), though the stabilizing terms are ultimately different in form.
In philosophy, the new methods can be seen as a generalization of the ideas present
for the PGEM methods for the Pi/P0 element, but significant differences appear as
well:
the linear interpolation of pressure chosen here requires extra care in order to
control the gradient of pressure in the formulation;
terms involving the jumps have a different coefficient than is present in the
PGEM for Px/Po.
The outline of this work is as follows: After some introductory material, we de
rive two general RELP methods in Sections 2.22.4 using an enrichment approach.
The methods are general in the sense that a total of four (related) methods arise from
them: two of the methods have enrichments which are analytically defined and so
lead to methods which are fully discretized (see (2.65) and (2.68)); the other two
require an extra approximation locally and so are derived as twolevel methods (see
(2.72) and (2.74), or (4.17) and (4.14)). Under the assumption no approximation at
the second level is required (i.e., all terms in the method are known exactly), the com
mon framework used in the derivation of the four methods is employed to show the
methods have unique solutions (see Lemmas 2.15 and 2.17), as well as to provide an
analysis for all methods. This analysis shows the methods to be optimally convergent
(see Theorems 3.15, 3.20, 3.27, 3.29). In addition to this analysis, the analysis for
the twolevel methods is considered, and all convergence properties are inherited for
this version (see Theorems 4.13, 4.16, 4.18, and 4.19). It is interesting to note that
6
the methods employing the Pi/Pfac element yield a locally mass conservative ve
locity, an important feature used when showing the convergence of a velocity in the
H(div, D.) norm (see Theorems 3.26, 3.31,4.17, and 4.21). Finally, we will consider
numerical tests which validate the theory in Chapter 5. We also explore performance
on the 5spot problem in the case of both a homogeneous and a heterogeneous porous
medium.
The rest of this introductory chapter is devoted to: 1) describing the Darcy model
in a general, and then more simplified form, as well as recalling the results used to
establish the existence of a unique solution, 2) introducing notation for the Darcy
problem and for discretization, 3) and closing with a few numerical methods for solv
ing the mixed form of the Darcy problem.
1.1 Preliminary notation
Assume D is an arbitrary bounded set in Kd. We will denote by L2(D) the usual
space of square integrable functions over D. For functions v, w G L2(D), we define
the inner product (; ')d and   D by
We shall use Ll(D) to denote functions belonging to L2(D) with zero average in
D (i.e., JDq = 0 if q 6 L\(D)). We also have a need for the space [L2(D)]2 =
L2{D) x L2(D) with inner product and norm defined by
{v,w)D:= vw, \vfD:= {v,v)D. (1.2)
Jd
Given multiindex a, the inner products for the spaces HS(D), which contains func
tions in L2(D) with all derivatives of order 0 < a < s in L2(D), will be denoted by
(1.1)
7
(, ),d, and the induced norms by fl s D. For v,w E HS(D), we take these to have
definitions
Furthermore, we define the space H(div, D) := {v e [L2(D)\2 : V v L2(D)},
and we define the associated inner product and the norm by
The space H0{div, D) stands for the space of functions belonging to H(div, D) which
have normal component vanishing on dD. We also recall the diameter of D is defined
by
where   is the Euclidean norm in Md.
Unless otherwise specified, Q denotes an open bounded domain in R2 with polyg
onal boundary dCl, and x = (xi, X2) is a typical point in Q.
1.2 The Problem
The goal of this work is to develop stabilized methods for the Darcy problem
and numerically test these methods both for consistency with the theoretical results
to be proved herein, and for performance on a realworld problem. In this section
we introduce the problem (1.7),(1.8). We then cast Darcys problem in a weak form
(1.12) and show that this formulation is wellposed in Theorem 1.12.
1.2.1 The Darcy Problem
K w)div,D := (v, w)D + (V w, V w)D, M2divD := H2d + IIV vfD.
diamD = sup{x y\ : x,y G D},
8
The Darcy problem is a set of equations relating a fluids velocity to pressure,
gravitational, and source terms. First, Darcys law describes the relationship between
the filtration velocity u (we simply refer to this as the velocity) and the gradient of
pressure p for a homogeneous porous medium.
(We have adopted / here as a term related to gravity to preserve notational consis
tency with [5, 6, 2]). This relationship has been put on a theoretical footing by Bear
[9] via upscaling techniques applied to the conservation of momentum equation for a
liquid phase. In (1.3), p denotes the viscosity of the fluid and k indicates the perme
ability of the porous medium. This is in general a secondorder tensor which arises
via an upscaling procedure to account for our incomplete knowledge of the exact
geometry of a porous medium.
Darcys law is supplemented with the conservation of mass, which may be written
(under the assumptions that the medium is rigid, homogeneous, and the liquid phase
is incompressible) as:
g indicating source terms. Defining 1C := (called the hydraulic conductivity),
Darcys problem on a domain Cl is formed using these last two equations:
u = k(Vp /).
ft
(1.3)
V u = g,
(1.4)
u I K,(Vp f) = 0 in D,
V u = g on Cl.
(1.5)
The general boundary condition for this problem is
au n + Pp b.
(1.6)
9
In the case that a 1 and [3 = 0, we are imposing Neumann boundary conditions
which fix the flux, and when a = 0 and (3 = 1, we are imposing Dirichlet boundary
conditions which fix the pressure. For our purposes, we will assume the former,
instead fixing the constant for the pressure solution by requiring fQ p = 0.
1.2.2 Simplifying assumptions
Assuming that the porous medium in question is homogeneous indicates /C is
constant throughout the domain. Next, under the assumption that the medium is
isotropic, we may write
(I indicating the identity tensor), with k now indicating permeability (see [9, 20]).
With a := ^ G K+ constant in the domain f2, the Darcy problem may now be written
as: Find u and p such that
Here, we have assumed homogeneous boundary conditions and that g Ã‚Â£ L2(Q) satis
fies the compatibility condition fS} g = 0. We make the further simplification that /
is a piecewise constant function.
Remark 1.1 Although we will assume, for the sake of simplicity, that u n 0,
we could more generally assume u n = b on dfl, and require satisfaction of the
compatibility condition JKg = b. In [27], it is shown that 3 Wb Ã‚Â£ H(div,Q)
such that Wb n = b. As such, we can recover the homogeneous case given above
with the modified righthand sides f owt and g V wb.
p p
ou + Vp f, V u = g in
(1.7)
u n = 0 on dQ,.
(1.8)
10
1.2.3 Weak form of Darcys problem
Darcys problem shows the relationship that must exist between velocity and
pressure in the case of a continuous velocity and pressure which is continuous with
continuous first derivative. However, it is wellknown that the corresponding function
spaces are not complete and therefore do not provide the correct setting in which to
provide existence and uniqueness results. To approach this, we must turn to the varia
tional form of (1.7). Formally dotting the first equation of (1.7) with v G H0(div, Q)
and multiplying the second equation of (1.7) by q G Lq(D) and integrating over
yields:
From here, we may integrate (1.9) by parts and apply the condition v n = 0 on <9fi
to yield the equivalent statement
Noting the boundary condition u n = 0, as well as the definitions (1.1) and
(1.2) of the L2inner products, we state a variational form of (1.7). Find (u,p) G
Ho(div, Q) x Lq(O) such that
(1.10)
(au,v)n (V v,p)n = {f,v)n Vug H0(div,Q), (1.11)
(V u, q)u = (g, q)a Vq G Ljj(n).
11
Alternatively, this can be expressed as: Find (u,p) H0(div, f2) x Lg(fi) such that
B((u,p),(v,q)) = F(v,q) V(v,q) Ã‚Â£ H0(div,Q) x (1.12)
where
B((u,p),(v,q)) := (
F(v,q) := (f,v)ti (g,q)si
(1.14)
We shall frequently refer to the symmetric form B(, ).
A fundamental question we must address when trying to develop a numerical
method to approximate the solution of a problem is whether the problem has a so
lution to approximate in the first place. Making use of classical theory, the results
of which are listed in the appendix, we state the wellposedness of the weak Darcy
problem in (1.12).
Theorem 1.2 Problem (1.12) is wellposed when we assume f2 is starshaped (see
Remark B.5). Moreover, defining a(u, v) := (ou, v), the a priori estimates hold:
Proof: First, note that a(u,v) = (ou,v)n is coercive on Ker(V) C H0(div,Q)
by definition of the [L2(Q)]2 norm. So, by Lemma B.3, there exists a > 0 such that
conditions (B.2) are satisfied. Next, since we may write [//o(fi)]d C H0(div,Ã‚Â£l), it
follows from Theorem B.4 that operator V : H0(div, fi) > L^fl) is surjective. But
this implies that there exists ft > 0 such that condition (B.3) is satisfied. So, we have
proved the conditions of Theorem B.l, and so problem (1.12) is wellposed, and the
a priori estimates follow immediately by observing a = a.
(1.15)
(1.16)
12
1.3 Discretization and approximation
Although Theorem 1.2 assures us of a theoretical solution to the Darcy problem,
it is not always clear what this solution is. We therefore turn to an approximation
approach based on finite elements in order to find an approximate solution. This
requires finitedimensional spaces defined on a triangulation of the domain Q. In
Subsection 1.3.1, we first focus on forming these spaces and the related notation.
With these points established, we define the main problem we address in this work
together with a few alternatives for dealing with it in Subsection 1.3.2.
1.3.1 Notation related to discretization
Let {Th} be a family of regular partitions TH of composed of triangles K.
Define
K0 := {u G L2(n) : v\K G F0(K) VK G T}, (1.17)
Vk := {v e C(Q) : v\K G P k(K)VK TH,k> 1}. (1.18)
We set Vfc := [V*]2 D H0(div, f2). Next, define
Qfc := Vfc n Lg(fi), (1.19)
Q_* := {q G L2(Q) : q\K G Pk{K)VK G TH}, (1.20)
the spaces of piecewisedefined polynomials which are continuous (resp. discontin
uous) over Q. It will prove convenient in the following chapters to refer to these last
two spaces collectively when linear interpolation is used. In this case, we refer to
them as Qd, d G {1,1}.
Denote by dK the boundary of K G TH which consists of edges F. Denote by
Ã‚Â£h the set of internal edges of 7The area of K will be denoted by \K\. We define
13
n
K
Fi
K
lie" np
*j 'j
K
F.
Figure 1.2: The normal vector.
Hf := diamF, HK := diamF, and H := max{HK : K e TH}. It is clear that
\K\ < HAlso, for each F = K n K' g Ã‚Â£h (see Figure 1.2) we choose a fixed unit
normal vector nF. The standard outward normal vector at the edge F with respect to
the element K is denoted by Tip, and coincides with nF in the case F c <9fi.
Define
Hi(T) := {veL2(Q) : :v\KeHl(K)}, (1.21)
L20(Th) := {v L2(Q) : :v\KEL2(K)}, (1.22)
H0(div,TH) := {veL2{n) : :v\KEH0(div,K)}. (1.23)
Several operators acting on these spaces will be used in the discussion. Let D be
some domain in Rd, d = 1,2, and suppose Z is a vector space defined on D. We shall
denote the identity operator by Ip Given a scalar function q, its projection nÃ‚Â£(r/)
into the Hilbert space Z defined on domain D will be given by (IIo(q), z)D = (q, z)D
14
Vz Ã‚Â£ Z. When D = Q, and Z has a definition on all of fi, we shall take 11^(q) =
Yh
IlD(g) := JDq. Borrowing from [3], one denotes its jump [7] and average value
{9} as
M := q\Kn$ + q\Kn%', {<7} := ^(q\K + q\K.),
respectively. We may further define the jump and average values of a vectorial quan
tity v by
H := V\k F + V\k' nFi M : = ^ (V\k + V\k')
Suppose q is a function for which we can define its jump across an edge F e Sh,
and let K, K1 eTH be the two triangles incident to F. Then, for arbitrary integrable
vector v, it follows,
(v, M)f = J v[q] = J (v n^qK + v n^'qK'^ = (v nÃ‚Â£, [gj n$)F.
(1.24)
From this, we can clearly see
M\f = M nF\2F, (1.25)
and so we shall interchange these expressions of the norm wherever necessary.
Furthermore, the following identity holds (see, e.g., [3])
^2(q,v n)dK = ^2 flfoi MV + ^2 (M MV, 0 26>
KT FeÃ‚Â£H FeÃ‚Â£H
when we define [] = 0 and {<7} = 0 on dfl. Also, take [<7] = 0 on dQ.
15
Finally, two local operators nK() and Mk{) acting on [Pi (A')]2 are given by the
expressions
kk(vi) := ^2 ^F{vvrip)(pp Nk(vi) := ^ IlF{vv nF)r)F, (1.27)
FcdK FcdK
where vx G [Pi(A')]2. The function
being the vertex of cell K opposite to the edge F. These form the local basis of
the lowest order RaviartThomas space, and we see that nK(vi) is the local Raviart
Thomas interpolantofui. Furthermore, we define the operator 7r( ) acting on Vi such
that 7r(ui)* = kk(vi). In accordance with the relevant definitions, we conclude that
7r(i) G H0(div, fl) is the global RaviartThomas interpolant of V\ (see [34, 4, 22]).
The functions r)F are elements of Ll(K) with the property Vr)F = oipF, and are
given by the expression:
'? = _Wf(^_x?c + CF) '29)
where CF is fixed to ensure Jkt]f =0 (see [5, 6]).
Remark 1.3 It may be shown that the operator n() is injective.
1.3.2 Numerical approximations
Consider the the unstable mixed method (i.e., that which leads to a singular stiff
ness matrix) for the Darcy problem when discretization is carried out with the Pi/Pi
element for both continuous and discontinuous pressures. Find (ui,pi) G Vi x Qj
such that
B((ui,pi),(i,<7i)) = F(i>i,<7i) (1.30)
16
2
10
10"
0 200 400 600 800 1000 1200
Figure 1.3: The lefthand plot shows how the singular values are distributed, with
the xaxis showing the values (no meaning is assigned to the yaxis). The righthand
plot shows the value of the singular values along the yaxis. In this case, the xaxis
indicates the position of the singular values in a vector ordering them from largest to
smallest, with a total of 1023.
for all (vi,qi) G Vi x Qj. Figure 1.3 displays plots of the singular values of the
stiffness matrix formed from method 1.30 using discontinuous pressure interpolation
for a particular instance of the crisscross mesh, similar to that shown in Figure 1.4.
Of the 1,023 singular values associated with the stiffness matrix, we observe a large
portion are numerically zero, indicating a large nullspace. In order to implement the
the associated elements for the Darcy problem, it is essential to handle this nontrivial
nullspace.
Assuming an efficient description of a basis for the nullspace, one possible ap
proach to finding a solution to the method could be to solve the resulting system in
17
the orthogonal complement to the nullspace.
A different approach to handling the nullspace is to stabilize the method through
the addition of extra terms. Approaches which follow this philosophy are called stabi
lized finite element methods. The methods in this work can be seen as an instance of
the class of stabilized finite element methods, and so we present some related meth
ods here for completeness. In [29], this formulation (among all other possible orders
of discretization for velocity and pressure) was stabilized with the terms
\ Ã‚Â£ ((^u1 + Vp1),^{av1 + Vql)) + Ã‚Â£ ( if I biUgil") (131)
2^V * )k Jf
Here, /? is a positive constant. Also, H and (a) are defined as,
(l,2)
where a is the value of a as seen from K and a' is the value as seen from K' (for
reference, see Figure 1.2).
Furthermore, in the case of a continuous pressure interpolation, the method in
[11] stabilizes the formulation (again, among other choices for interpolation) with
the leastsquares term
j2 (Pi n0(pi),gi n0(<7i))n, (1.33)
where projection n0 indicates the projection into the space of piecewise discontinuous
constants on the domain Q, a is a positive constant, and L is a characteristic length.
Finally, we note one of the PGEM methods designed to stabilize the Pi/Po ele
ment for the Darcy problem (1.7) (see [6]) is: Find (mi p0) 6 Vx x Q0 such that
B((7r(uI)>pd),Wi),9b)) Ã‚Â£ ~~"(IPdIj [?o])f = F(7r(i), q0) 034)
FeÃ‚Â£H
18
for all {vuq0) G Vi x Q0.
This method was derived by enriching the velocity and pressure spaces, and in
dicates an enriched pressure solution,
Ph '= Po + 51 7lF ( ~IPol nF + nF(ui nF) J . (1.35)
FeÃ‚Â£ a '
Here, r)F is defined in terms of t)f (1.29) by:
Pf(x)
nF, x G K and F C dK
0, x G K and F Ã‚Â£ dK.
Numerical experiments have validated the expected convergence properties of the
method. Moreover, we have seen that for the family of meshes shown in Figure 1.4,
pressure ph is actually superconvergent (in the sense that we see a higher rate of
convergence than expected with the piecewise constant interpolated pressure without
increasing the number of degrees of freedom in the problem).
Due to the nice properties of the PGEM for the Pi/P0 element, the following
work focuses on extending the underlying philosophy to the Pi/Pi element for both
continuous and discontinuous pressure interpolations. This will be done borrow
ing from the notion of consistently stabilizing as in [29], using terms which involve
projections into piecewise discontinuous spaces as in [11], all within an enrichment
framework driven by residuals as in, e.g. [6].
Remark 1.4 A small note regarding the term stability should be presented. The
use of the word "stable" is overloaded in the framework of finite element methods.
In some cases, the term refers to the sensitivity of the method with respect to the
data controlling the relative size of certain terms present in the method. In limiting
cases for these data, the methods may yield approximate solutions which exhibit non
19
Figure 1.4: A sample crisscross mesh and a curve showing better than 0(H2) con
vergence for pH in the L2norm on a family of this type of mesh.
physical oscillations arising from the fact that the system matrix is nearly numerically
singular. In other cases, such as with mixed methods, the term stable may refer to
the invertibility of the matrix when formed using different combinations of approx
imation spaces to discretize the fields which are to be approximated. In this case,
instability is not numerical in nature, but reflects a fundamental incompatibility of
the spaces as described by the famous LBB condition (see, e.g., [16]). This latter
form of instability is the one we address in this work.
20
2. The RELP methods
In this chapter, we introduce several methods by which we obtain approximate
solutions of the Darcy problem. The methods all rely upon the same strategy for
their development, and as such are derived in a common framework. In Chapter
3, we shall see that this new approach to deriving the methods will provide a good
basis for performing a numerical analysis of the methods. In a general sense, this
framework begins with an enrichment of the linear spaces for approximating both
the velocity and pressure, and we derive the methods with the philosophy that the
enrichment functions solve local problems driven by local projections of residuals.
For this reason, we call these methods Residual Local Projection, or RELP, methods.
The outline of this chapter is as follows:
Preliminary definitions and results are introduced in Section 2.1.
Using an enrichment of the polynomial spaces composed of linear velocities
and pressures, two general methods are derived in the course of sections 2.2
2.4. In particular, we will motivate the need for, and requirements on, the
enriching spaces in Section 2.2. In Section 2.3, we will use these motivations
to explicitly define these spaces. In Section 2.4, these spaces are used to derive
the methods, which are presented in Subsection 2.4.3.
The two general methods are shown to have unique solutions in Section 2.5.
In Section 2.6, we derive four methods based upon the general methods of
Subsection 2.4.3.
21
2.1 Preliminaries
We will make use of several nonstandard operators and spaces in the sequel, and
we focus on their definitions in this section. First, a set of operators acting on the
space Vi will be defined in Subsection 2.1.1. We will then define a few subspaces
of H0(div, f2) and Lq(f2) in Subsection 2.1.2. Lemma 2.2 of this subsection will be
important to the derivation presented in Section 2.4.
2.1.1 A few operators
A set of operators will be required throughout this work. We begin with their
definitions and establish a few properties relating the spaces we will be considering.
First, denote by uF{v nF) the operator which stands for either IF(v rip)
v rip or Il/r(V rip). We make use of this to define operators pp() and Vp() acting
on functions V\ Ã‚Â£ [P^/f)]2 and having values in H(div, K) and Ll(K), respectively:
i, (2.1)
pK(vi)np = i/F(i nF).
We see first of all that the functions pK{vi) defined in this way satisfy the compatibil
ity condition JK V pk(vi) = fgK Pk(vi) nK, and so problem (2.1) is wellposed.
Global versions p() and V() acting on Vj Vi are defined such that p{v\)\x =
Pk(vi) and V{v\)\k = Vk(vi). It is clear that p(vj) e H0(div, Q) as p(vi) shares
with v\ the continuity of their normal components.
We make a few important observations regarding problem (2.1). First, if
i/p(vinF) = nF(vvnF), (2.2)
22
it is easy to verify [5, 2, 6] that
Pk(vi) = nK(vi), Vk(vi)=J^k{vi), (2.3)
(where itF and J\fK are defined in (1.27)), and we conclude that (2.1) has an ana
lytical solution in this case. On the other hand, a numerical method is required to
approximate the solution to (2.1), if
uF(vv nF) = vy nF. (2.4)
For the time being, we leave the strict definition of vF alone. However, the choice of
either (2.2) or (2.4) will ultimately lead to one of two branches of numerical methods.
This will be explored in more detail in the sequel.
Remark 2.1 It may be shown that when Pk{') Pi(A) > pk(Fi(K)) satisfies the
boundary condition Pk{v\) n v\ n, then it is injective, and we conclude that the
global operator p() is injective as well. See Lemma 4.3.
2.1.2 A few subspaces
As has been indicated, we will derive methods based on residuals. We will fix
notation related to these residuals below. Given arbitrary functions G Vi and
problem,
RM(vi,qi,f)^f~crviVqi and Rc{vu g) := g V t7j. (2.5)
We would like to define a subspace of Lg(f2) base upon RM(vi,qi, f). Its definition
will require the following problem of finding q L^(K) WK G TH such that,
Aq = VRM(vuqi,f) in K, (2.6)
Vq nF = isf(Rm(vi,qi, /)) nF on F C dK.
23
Noting that JK Aq = fdK \7q n, problem (2.6) is wellposed.
Letting wF(vi,qi, f) denote the formal inversion operator for this problem,
we also consider a global form w(v\,qi,f) of this operator with the property
w{vi,qi, f)\F = u)k{v\, qi, /). Assuming / is fixed, we define the following finite
dimensional space:
W := {q H2(Th) n L\(Th) : q = w(vi,quf) Vvj G Vi,qi G Qd}
Rather, we see W is composed of all solutions to problem (2.6). We denote the
orthogonal complement of W in Ll(Th) by W1.
Let p(Vi) to be the image of the space Vi under the operator p(). The next
Lemma presents important information regarding the relationship between some of
the function spaces we have defined.
Lemma 2.2
Q
Proof: First, suppose q G Qd fl W1. Then (q, t) = 0 Vi G W. Next, select t e W
such that 1\k = wk(0,7,0), rather, t satisfies the local problems for each K G Th,
At = V Rm{0, q, 0) = 0 in K, (2.8)
Vi nF = nF(/?M(0, q, 0)) nF = S7q nF on F C dK. (2.9)
The unique solution i to this problem, since q G W1 C L\{Th), has the property
t\K = q\K ~ IM?) = q\K It follows then that 0 = (q,t) = Mk and so
9 0.
24
Next, suppose v p(Vi) fl H0(div,TH). Since v 6 p(Vi), it follows that there
exists V\ Ã‚Â£ Vi such that v = p(vi) and V{v\) satisfy the local problems
Pk(v i) + Wk(vi) = 0, = VUik, (2.10)
Pk(vi) nF = !/(! nF).
Since v Ã‚Â£ H0(div, TH), v nF = p(vi) nF = 0 for each F Ã‚Â£h Moreover, since
V v\k = V pk{vi) = V Vi\k is constant, the divergence theorem applied on each
K lTh tells us:
Vv]k=W\LVv
=w\LvnK
= 0.
So, in fact (2.10) has zero righthand sides, indicating v p{v\) = 0.
2.2 The enriching approach: Motivation
As the linear spaces defined in the previous section have the properties:
Vi C H0{div,n), Qd C Ll(Q), (2.11)
a natural starting point for approximating the solutions (u,p) to problem (1.12) is:
Find (ui,pi) 6 VL x such that
B((ui,pi),(t7i,<7i)) = F (!,?!), (2.12)
for all V\ Vi and q\ Qd Unfortunately, it is wellknown that
inf
91 &Qd
(Vt>i,gi)n
mVi ll*7illo,nlvi
= 0,
(2.13)
and it is clear that condition (B.3) of Theorem B.1 is not satisfied. In this case, the
issue is that the space Vj is not large enough for V: V] Qd to be surjective.
25
Remark 23 See [22] for an explanation of the famous checkerboard mode, which
shows the result in Equation (2.13).
Our goal is to derive methods which overcome this incompatibility of spaces by sup
plementing the linear spaces of problem (2.12) with spaces of functions that contain
the information necessary to satisfy the LBB condition.
We will derive the methods based upon the PGEM approach [2, 6, 5]. To begin,
we assume trial spaces which are as large as possible. Thus, we choose the trial
spaces
V:= Vi + H0(div,n)
and
Q := Qd + Lg(fi).
(Note, we could simply write H0(div, Q) and Ll(Q), but we choose the above nota
tion to emphasize that we want our trial functions to retain information about degrees
of freedom related to V! and Qd) Ultimately, we will choose suitable subspaces of
these trial spaces. For now, consider the following multiscale decomposition of the
functions in these spaces: write (//,?//) = (vi I ve,qi + qe), where Vi Vj,
ve H0(div,Q), qi Qd, and qe e Lq(Q). As for the test space, we will take
it to be as large and as local as possible such that (vn, qu) decompose uniquely as
(v\ + Vb, qi + qb) With this in mind, we choose for the velocity and pressure test
spaces, respectively:
p(Vi)HQ(div,TH), Qd(BW. (2.14)
This choice is motivated by Lemma 2.2, a result which will allow us to use the static
condensation procedure. We shall consistently refer to ue, pe, and pb as enrich
26
ment functions.
We now modify problem (2.12) using these updated spaces and define the fol
lowing PetrovGalerkin method: Find (uh,Ph) G V x Q such that
(vH, qff)) = F(vff,qff), (2.15)
for all vH = p{v i) + vbÃ‚Â£ p(V i) H0(div,TH) and qH = qi + qb G Qd ^X
We must begin by asking the important question as to whether the problem (2.15)
is wellposed. In fact, the answer up to this point is no. However, we investigate the
reason for this in the sequel and rectify the problems using the new enriching strategy.
We refer to Theorems B.6 and B.7 in the appendix to provide the framework needed
to investigate this issue. We may write (2.15) as: Find (uh,Ph) G V x Q such that
(cTUH,vH)n (VvH,pHh = {f,vH)n VvH e p(Vi) H0{div,TH),
(V Uh, 9//)o = (<7i <7//)sj V<7// G Qd W. (2.16)
Defining C := V: p(Vj) 0 H0(div,TH)  Qd + Lg(T), B := V: V, +
H0(div,Tf{) * Qd W1, and A I : Ker(B) > Ker(C), we see this fits
the framework considered in Theorem B.6. There are two things we need to note.
CT is not injective since Qd + Ll(TH) is too large. Consider the related term
(Vi7 H,pH)n = (Vt>i,pi)n + (Vv6,pi)n + (Vvb,pe)n,
where we have used the constant value of V Vi to establish (V yi,pe)n 0.
First, since V: H0(div, K) Lq(K) is surjective, it follows that the last term
on the righthand side above controls pe on each K, and this indicates that pe
is controlled on all of fi. We now question what control over pt exists. As
27
previously noted, Vj and Qj are incompatible spaces, so we do not expect to
gain control overpj with the term (VUi,pi)o. In fact, we may see where the
deficiency arises by considering that there exist positive constants C\ and C2
such that (See [26] for details),
{VVi,qi)n
Mi
> Cjfo
\h
^Vgi^ +
Ilo./C
(2.17)
So, in order to control p\, we must control Vpi and pil Therefore, the re
sponsibility of controlling these values falls to vb through the term (V vb, p\)u
However, integration by parts and the fact that vb nF = 0 for each F G Ã‚Â£//
indicates
(Vvb,Pi)K = 
KTh KTh L
= XI vpi)*
{vb,Vpx)K+ ^2 (vbn^,px)F
FCdK
(2.18)
KerH
This makes clear that the term (V vb,px)n accounts for the deficiency in infor
mation regarding Vpi because for each K Ã‚Â£ TH there is a vb\F Ã‚Â£ H0(div, K)
such that VV(,k = ^t{pi\k ^(pi)), meaning that substitution and the
inverse inequality imply
(vi7tlpi)jr = (vt>6,p1nK(pi))K =  uk(pi)Ik > ^WpAk
(2.19)
However, the term cannot account for the deficiency in information regarding
bil
A is not an isomorphism since H0(div, f2) (and importantly, Ker(B)) is too
large.
28
Remark 2.4 Problem (2.16) indicates in particular that (V U//, qb)n = (g, qb)nfor
all qb Ã‚Â£ W1. Rather, (V ue,qb)K {Ff ,qb)K for all qb Ã‚Â£ W1 and K Ã‚Â£ TH,
leading us to conclude nJ^x(V ue) (Rc).
We see that in order to have a wellposed problem, we need to restrict the func
tions ue and pe to subspaces of H0(div, Q) and Ljj(fi), respectively. We motivate the
choice of these subspaces with the following points:
1. Define subspaces of the trial spaces we have chosen in problem (2.15) for which
the formulation is conforming and wellposed;
2. Yield enrichment solutions which are residualbased.
3. Dont increase the number of degrees of freedom over those present in the
residuals.
2.3 The enriching approach: Defining appropriate subspaces
In this section, points 13 at the end of the last section will be addressed in terms
of the discussion related to operators C and
enforcing ue Ã‚Â£ H0(div, TF) (rather, enforcing zeroflux boundary conditions for
functions ue on the edges of the triangulation), just as is done for the enriching por
tions of the velocity test functions, will ensure that 4>A is an isomorphism. However,
we do not want to enforce ue nF = 0 on the edges F since doing so will destroy
the surjectivity of operator B through a loss of control over the jumps of q\. Instead,
with an eye on keeping the method conforming and residual based, we enforce for
each F Ã‚Â£ Eh
oue nF = (RM(uupuf) i'F(RM(uupuf))) nF + 1^nF([[pi])nF, (2.20)
tiF
29
where a > 0 is some real number independent of Hp and a. This choice is equivalent
to imposing homogeneous boundary conditions since nonzero boundary conditions
are accounted for with a perturbed righthand side when solving for a function in
H0{div, Th) (See the remark after problem (1.7)(l .8)). Enforcing ^Ilfdp!]]) nF
will replace into the method the information lost according to the comment on the
nontrivial kernel of CT. We hope this choice renders problem (2.15) wellposed, up
to V ue W1.
Note that Remark 2.4 indicates that nH/1(Vue) = nvv (Rc), but doesnt re
solve the value beyond this. So, we are permitted (and required) to choose the value
of Illv+R(V ue). This motivates our choice for the subspace of H0(div, Q) we seek.
Again, with an eye on stabilizing the method using residuals, and taking care to ensure
the method is conforming and that the compatibility condition fKV ue JgK ue nK
is satisfied, we choose ue to come from the space:
Ve := {v H0(div,n) : IlÃ‚Â£+R(V v)\K = JK(RM, Rc, foil)}, (2.21)
where
Jk(RM,RCM) = ^Tw(vuql,f) + n^(Rc(v1,g))UK(Rc(vug)) + CK.
Here, Ck is a constant meant to ensure compatibility with the boundary condition
^7FI/r([<7i]]) nF and has definition,
c*=: E (222)
FCdK ' '
30
Having addressed the points regarding C and
now take for the enrichment pressure space
Qe := Ll(T). (2.23)
With these definitions we have determined the spaces in which we seek a solution
to problem (2.15). We shall derive the numerical methods to be considered here and
show that they are wellposed.
2.4 The enriching approach: Derivation of the methods
The problem we now wish to consider is: Find (uh,Ph) E (Vi+Ve)x(Qd+Qe)
such that
B({uH,pH), {vH,qH)) = F(vH,qH), (2.24)
for all vH = p(v i) + vbe p(V i) V6 and qH = qi + qb e Qd ^/
Consider the problem in an equivalent form, which we obtain through a standard
static condensation procedure:
B((ui + ue,pl),(p(vl),qi) = F(p(v1),q1) V(p(wi),^) 6 p(Vi) x Qd, (2.25)
(ouH,vb)K (pH,Vvb)K {qb,VuH)K = (f,vb)K (g, qb)K (2.26)
V(vb,%) E H0(div,TH) x iyx.
Problem (2.24) is wellposed if and only if the system (2.25), (2.26) is well
posed. In this system, (2.25) represents the method and (2.26) represents local prob
lems. This section will focus on three points related to system (2.25) and (2.26). First,
we will discuss the structure of the local problems and develop a notion of how their
solutions will affect the method. We then describe solutions to the local problems.
31
Finally, we will firmly define two general methods by substituting the local solutions
into the (2.25).
2.4.1 Local problems in the method
The local problems may be written as:
&K((ue,pe),(vb,qb)) = (RM(u1,pi,f),vb)K(R?(ui,g),qb)K. (2.27)
From now on, when it is clear that RM(wi,ri,f) and Rc(w\,g) depend on the
solutions (ui, pi) to method (2.25), we shall shorten the notation of these to RM and
Rc, respectively.
Using the definition of Ve, this local problem, under the appropriate assumptions,
is equivalent to the following strong local problem:
aue + Vpe = Rm VuÃ‚Â£ = Jk{RM,RCM) + ^W\RC) intf,
(2.28)
aue nF = ^Rm uf{Rm) + tif on F C dK ,
for each K gTh Motivated by the linearity of this problem, we define the functions
v%{, vf, and uf by the problems,
a Vg1 + V9eM = Rm(vu quf), V v? = 0 in K , (2.29)
ov nF = (RM{vuquf) v>F{RM(vuquf)))nF on F C dK,
a V' + Vq^ = 0 in K
Vvf = Rc(vl,g) UK(Rc{v1,g)) 9i> /) in K ,
nF 0 on F C dK, (2.30)
32
and,
f + V7f = 0, Vf = Tir E nF([gi])n?,
v? nF = 7^nF(M) nF on F C dK. (2.31)
tip a
Then we see the solutions (ue, Pe) of problem (2.28) satisfy ue = + uf and
Pe = P + Pe + Pe where (u^^pM), (uf ,pf), (uf ,pf) satisfy problems (2.29),
(2.30), and (2.31) with Vi 3 Vi = U\ and Qd^ qi = Pi. respectively.
Remark 2.5 It is important to remark that the conditions on ue rip satisfy the com
patibility condition given by the divergence theorem. First, we see that either of the
possible two values for Vp(RM) satisfy the compatibility condition for u^1. Indeed,
for either choice:
f u"n= Ã‚Â£ f (Rm vp{RM)).nF
JdK FcdK ^F
= E / (RMMRM))np
FcdK F
FcdK
= 0
= f Vu".
Jk
Also, u satisfies the compatibility condition:
W\L>m)'nf = ^Ã‚Â£JrU'n
= M/,*"
= cK.
(2.32)
33
Finally, since w(uup\, f )\K g 1%(K), it is clear the compatibility condition is sat
isfied in problem (2.30).
Remark 2.6 It is important to note that (ue,pe) (and also (u,Pe), (u^1 ,p^), and
are defined globally in H0(div,Q) x L\{Th) through the local problem
(2.28), which they satisfy for each K G TH. This follows because problem (2.20)
ensures [tte] = [uf ] = [uf ] = [uf ]] = 0.
It is now clear that problems (2.29)(2.31) are wellposed and define an enrich
ment solution (ue,pe) to (2.27) in terms of the linear portion of the solution (ui,pi).
Having considered the significance of local problem (2.26) we now consider
method (2.25), which may be rewritten as
i(ue +u?p(vi))k ~ (V(u^ + u?),ql)K]
kcth
+B((uj ,pi),(p(vi),gi)) = F(p(t71),g1). (2.33)
We may rewrite (2.33) in an equivalent form. First, problem (2.1), integration by
parts, and using uf = 0 for each F G Ã‚Â£h gives us,
aptui))* = (uf, Wk(vi))k
= (Vu,Vk(vi))k,
and we conclude, using V G L\{K) for all K eTH,
^2 [{u^,op{vi))K ~ (V"U^,<7i)k]
kct
= ^(^^eMvi)ql+nK(ql))K. (2.34)
KT
34
Furthermore, integration by parts also shows us,
(V uf, qi)K = (uf, Vqi)K (uf nK, qx)BK,
and we conclude from this, the fact [uf ] = 0 on all internal edges, and the identity
(1.26),
X [(ttf.MVl))* (Vuf,^)*] = X (Ue^P(Vl) + Vq\)K
KTh KTh
 Ã‚Â£(txeD,nF([M))F. (2.35)
FÃ‚Â£h
Substituting terms (2.34) and (2.35) into equation (2.33), we see the method is
described by the equation,
B((i +u^',p1),(p(v1),q1) + ^ {VUe,VK(vi) q! +nK(qi))K (2.36)
KTh
+ X (uf ffp(vi) + V<7i)* X Uf nf(IM) uf)f = F(p(i), qi).
KTh FÃ‚Â£
Written in this form, it is clear an implementation of the method requires explicit
definitions for the functions uf and uf. However, we need only include V uf from
problem (2.30), for which we currently have all terms, aside from w(u\,p\, f). We
now turn to either giving analytic expressions for, or at least characterizing, these
enrichment solutions.
2.4.2 Solutions to the local problems
First, consider problem (2.31), which uf satisfies for each K 6 TH,
auf + Vpf = 0, Vuf = X nf(bll) nF
uf Uf = 7^nF([pi]) nF on FddK.
Hf <7
35
Denote the formal inversion operator of this problem by
(ue \K,Pe k) = 'Pa'(IPiD)
and let 2^([[pi]]) := p^\k and 'D^(piJ) := uf \k Following the approach taken in
u.
e\n,Pe k) to this problem is given by:
1 onF(pi!) K Ik 2^ Hfo Pf and FcdK * (2.37)
D, allpdpi]) nÃ‚Â£_K vAk Hra (2.38)
where ipF and ijp are given in (1.28) and (1.29), respectively.
Next, consider problem (2.29)
<7 u" + Vp" = Rm, V uf = 0 in K,
Denote the formal inversion operator of this problem by
K) = Mk(Rm) = MK{f
Due to the assumption that / is piecewise constant, as is Vpi, we conclude that
MK(f Vpx) = (0, (1K n*)(/ x p,)). (2.39)
We now address the portion of the solution (ii^\k,p^\k) = i). It is clear
that
<7tk" + Vp" = auu V u" = 0 in K, (2.40)
au^1 rip = au\ nF + uF{au\ nF) on dK.
Comparing to problem (2.1), we see thatp^ = Vk(ui) and u" pF(ui) Wi
36
Remark 2.7 In fact, define oMpK(ui) := p^\k and oMuk{ui) := u^'\k Then,
since V (oMpk(ui)) = o(ux oM^(ui)) =
that VK{ui) = aMpK(ui) = pf and pK(ui) = (1K aM^)(ux) = Ux + it.
Using this notation, we see the solution to problem (2.29) is
ue*lk = {pkTk){u\), P^\k Vk{u\) + {Ik Rt<)(fx~P\)\k (2.41)
Remark 2.8 In fact, this solution is only explicit up to the definition of the operator
Pk(u i). which is itself dependent on the explicit choice of boundary condition uF().
We discuss the implications of the choice momentarily.
We are not able to derive analytical solutions for uG, pG as defined in problem
(2.30). However, this is not an issue from the point of view of the implementation
of the method (2.36), as only VuG must be included. We know the value of this
from problem (2.30). If required for postprocessing, this may be approximated by
twolevel methods following [2]. We also note that
(Ul,Puf)=Pe (242)
This may be seen by noting that problem (2.29) implies p satisfies:
Apf = V.ftM, Vp^ nF = uf(Rm nF), (2.43)
and comparing to problem (2.6).
Taking the solutions to equations (2.29)(2.31) together, we conclude that
ek = (Pk ~ 1k){ux) + ug + Ã‚Â£>Ã‚Â£(IpiJ), (244)
pAk = VK(Ul) + (1K n*)(/ X Px)\k + pGe + (245)
37
2.4.3 Two general methods
Substituting the solutions for u^, and into (2.36) and noting the injec
tivity of the global operator p() (see Remarks 1.3 and 2.1), the method takes on the
form: Find (U{,pi) Vi x Qd such that
B9{(u1,p1),(vuql)) = F9(v1,ql), (2.46)
for all (ui,7i) G Vj x Qd, where,
B^Ui.pi),^!,?!)) := B((p(1),p1),(p(u1),g1))
+ E (Vk(IPiI),Pk(vi) + Vq1)K
kzth
~ E 772 {'PK{ui)pi+nK(p1),VK(vi)qi+IlK(ql))K
KT K
E
FeÃ‚Â£H
^(IIf(IIpiJ), nF([g1]))F,
Fs(vi,9i) := (f,p{vi))n
E (n*(s) +
k&t '
1
(iKnK)(fx),qi
(2.47)
K
 E (9Z7jTfx>pidvi)) (248)
KZT V UtiK ' K
We have now arrived at a final form for what shall hereafter be referred to as the
general method.
We also introduce a reduced form of this method. This reduced form is obtained
by neglecting the terms J2kTh(u?(Tp(v 1) + Vgi). We shall see that doing so
doesnt undermine the stability or convergence rates of the general finite element
38
method and has the advantage of yielding a method which is symmetric. Taking,
Bf((wi.Pi).(wi.9i)) '= B((p(ui),Pi),(p{v\),qi))
KT
FÃ‚Â£h
(2.49)
the method is: Find (ui,pi) Vi x Qd such that,
b?((i.Pi)>(i>9i)) = F3(i,gi), (2.50)
for all (vi,qi) E Vi x Qd. This method shall hereafter be referred to as the sym
metric general method.
Remark 2.9 Note that we have dropped the term
(V U\ 11*(V Ui),Vk{vi) q\ + I1a(<7i))a:,
because V U\ 11* (V U\) = 0. However, we continue to regard this term as present
in the methods. This is particularly important when the consistency of the methods is
proved in the next chapter.
Remark 2.10 Although we shall perform the error analysis for the methods as given
above, the term,
 (251)
KT
may be dropped when performing computations. In Section 3.4.3, this term is shown
to be at order II2, and as such neglecting it does not undermine the convergence
properties of the method.
39
2.5 Existence and Uniqueness Results
We must now show the methods (2.46) and (2.50) presented in Section 2.4 have
unique solutions. We begin by defining the meshdependent norm:
1
(.P)III// := *Min+ X lVPl
oHf
l!nF(bl)lo,F
1 1/2
(2.52)
KTh FzCh
The methods will be shown to satisfy an infsup condition in this norm in Subsec
tions 2.5.2 and 2.5.3. The theory of these two subsections requires a few preliminary
results, which can be found in Subsection 2.5.1.
2.5.1 Some auxiliary results
We now establish some auxiliary estimates on terms appearing in the method due
to enrichment. The overall goal of these results is to bound terms present in methods
(2.46) and (2.50). This will assist us in establishing the results of Subsections 2.5.2
and 2.5.3, as well as the convergence theory in the following chapters.
Lemma 2.11 Let K G Tw. Then there is a constant Ci from the inverse inequality
such that
^Hk\Vpi0iK < Pl IMpOIIo,* < VPl0j/o (2.53)
U/ 7T
and
I^K(.)lojt < P.54)
7T
Proof: The first of these results follows directly from the inverse inequality of
Lemma A.3 on the one hand, and the Poincare inequality A.2 on the other, noting
that pi 11* fo) G Lq(K). The second follows by observing Vk(ui) e Lq(K) and
applying the Poincard inequality.
40
In particular, result (2.53) of Lemma 2.11 establishes an equivalence between a
terms present in the methods and one of the key terms we seek to control according
to equation 2.17. Now, note the following lemma
Lemma 2.12 For each K e Th, let z G L2{dK) and constant Dk defined on
K satisfy the compatibility condition JK Dk = fgK z. Further, suppose (v, q) e
H(div, K) x [Lq(K) fl H2(K)\ satisfies the local problem
ov + Vg = 0, V v = Dk in K,
v Tip = z on dK.
Then there exist constants C\ and C2 independent of Hk and a such that
CxHxJ,2\z\dK < Vg* < C2aHll2\z\dK. (255)
Proof: From the definition of the L2 norm, integration by parts, the fact q L\(K),
CauchySchwarz inequality, and the local trace inequality (A.2), and the Poincard
inequality, we have
flV^ = (V(7,Vg)K
= (7, &q)k + (<7, Vg nK)dK
= o(q, V v)K + (g, Vg nK)dK
= (?, Vg nK)dK
V? n llo.a/c
/I \ 1/2
< Q + HkMIk) IIVg nK\QfiK
41
from which we conclude the second inequality of (2.55). To see the first inequality,
consider that the trace inequality (A.3) tells us
!o,a*: ll^7? nllo,9K
This establishes the first inequality of (2.55).
The previous lemma is useful in establishing a few bounds for Vgf, as in the
following corollary, which bounds its norm in terms of norms on its local boundary
conditions.
Corollary 2.13 Suppose q is the pressure solution given by problem (2.31). Then
there exist constants C3 and C4 independent of II k and a such that
Proof: We first note that qj? is a polynomial, so each of the results in Lemma 2.12
may be used. The Lemma implies, using mesh regularity,
Remark 2.14 We may define a global version of Corollary 2.13 by observing
The theorem follows from the definitions of C\ and C2 from Lemma 2.12.
(2.56)
KTh FcdK
FÃ‚Â£h
42
2.5.2 Existence of a unique solution for the symmetric general method
Lemma 2.15 Let B(.,.) be the bilinear form defined in (2.49). Then there exists a
positive constant (3, independent of H and o, such that
and we conclude that,
sup
(tui.rOeV, xQd{0} l(p(tOi),ri)H
> PI(p(vi),qi)lHi
for all (vuqi) G Vi x Qd.
Proof: Using the definition (2.49) of Bf and Lemma 2.11,
Bf((ui,gi),(vi,gi)) = (jb(vi)o,j2+ E T^lk lMgi)*
KTh k
+
FCÃ‚Â£h
FeÃ‚Â£u F KTh K
1
\Ik
>
sWi)il+ E ?^iv9io,
8' Cia
+ E ^lnH[9il)llo.F
FÃ‚Â£h F
?p(i)lin+ E 7k\v*\bc
8 K^h C,a
+ Y ^nF([M)L
where we set (3 = min{^y, }. The last result follows by noting that l(p(ui), qi)H =
H(p(i),9i)Hw.
43
Remark 2.16 Since the choice a = 1 is sufficient for the solution to exist, we will
make this assumption on the value of a whenever a symmetric method is considered.
2.5.3 Existence of a unique solution for the general method
Lemma 2.17 Let B9(.,.) be the bilinear form defined in (2.46). There exists C > 0
such that if a < C, then there exists a positive constant (3, independent of H and o,
such that
bs((vi,<7i),(ui,<7i)) > /?(p(i>i),9i)Ih,
and we conclude that.
sup
(tui.rOeVixQd{0}
B9((T?i,gi),(u>i,ri))
HMwiJ.rOI*
> Pi(p(vi),qi)lH,
for all (wi,gi) G Vi x Qd.
Proof: First, observe that we may write
B((wi,9i), (wi, gi)) = B?((wi,gi),(vi,g1)) E CPÃ‚Â£(M WM)*:
KT
~ E (^(M),V<7i)K. (2.58)
KTh
Equation (2.57) already establishes a bound for Bf((i,qi), (t>i, qi)), so we turn
to the other two terms on the righthand side. First, from the local problem (2.31),
Youngs inequality, and Corollary 2.13,
 E (WfalWfai))* = E (V9eD.P(Vl)k
KT KTh
> 
> 7i C4a E
FÃ‚Â£h
a
oH,
27i
P(i)2
44
where C4 = 4CTCf. Again, using problem (2.31), Youngs inequality and Corollary
2.13, we have,
 Ã‚Â£ (k(I
K
KTH
KTh
a Ã‚Â£ 2^1* Ã‚Â£
KeT
e IIK
KdTH
~L^k
^C*a E (259>
Fe Ã‚Â£h
Using these two bounds and Equation (2.57), we find there are positive constants
7i and 72 such that
Ã‚Â£>((., .), (i, ?,)) > (g ^) M,)2 + (2 i) Ã‚Â£ vi
+ (lC,a(7i+ft)) Ã‚Â£ (2.60)
F&Ã‚Â£H F
Taking 71 = 4/3, 72 = C/, and assuming a < 2Ct{l+Cry we obtain
Bs((vi,gi),(ui,<7i))>CT^p(ui)2 + ^ Vgi*
KeTH
+ (lG,a(i + C,)) Ã‚Â£ ^nF(W)U,F
' ' FÃ‚Â£Ã‚Â£h F
>5Ki)I2 + 5/ Ã‚Â£ v,i
1 KeTH
FÃ‚Â£Ã‚Â£h
>/?l(p(t;1),
where /3 = min{, ^7}. Noting that (p(t>i),gi)fltf = (p(vi)> <7i)lll//>the result
follows.
45
2.6 Explicit methods formed from the general methods
All that remains is to form explicit methods from the general methods by fixing
the boundary condition in the problem we solve for u = (p/c 2f)(u i) in problem
(2.29). This in turn requires us to fix the boundary condition for pk{u\) in problem
(2.1). As indicated previously, two choices are possible, and so a total of four methods
are presented. In the case of either choice, we note that whatever analysis we are
able to perform on the general and symmetric general methods that doesnt require
a specific boundary condition is inherited. In particular, the methods considered in
this section are wellposed owing to Lemmas 2.15 and 2.17. One choice, vF{au\) :=
Ilf (aiii), leads to a closed form for the function p(ut), and therefore a closed form
for u^1. Methods arising from this choice are presented in Subsection 2.6.1. For
the other, uF(aui) := aui, the analytic solution is not clear. Thus, we need to
approximate p(ui), leading us to develop a twolevel approach to solving the method.
These methods are presented in Subsection 2.6.2.
2.6.1 Methods with analyticallydefined terms
Assume that we want to solve problem (2.1) with iyF(auinF) = UF(auinF).
This yields the problem
Pk{u\) + VÃ‚Â£V(ui) = 0, Vptf(ui) = Vui, (2.62)
pK(ui)nF = UF(u1nF). (2.63)
We recall that (2.3) tells us pF(u\) = 7Tk(ui) and Vk{u\) = Nk{u\) (i.e., the local
RaviartThomas interpolant of U\ and its potential). From this, we conclude
uf \k = ttk(ui) till*, pfu =AfK(ui) + (1K UK)(f xpi)K. (2.64)
46
We see that the general method (2.46) takes the form: Find (ult pi) Vi x Qd such
that
B 1((tti,p1)T (Wi, ?!)) = (2.65)
for all (vuqi) Vi x Qd, where
B((ui,pi), (!,?!)) := B((7r(u1),p1),(7r(t71),gi))
+ X {Vk(IPiI),
>K
KeT
 X ST (^(tii) Pi + nK(Pi),AA:(ui) 9i + nK(gi))K
 X t77(iM[IpiJ)>iM[[<7i]))f,
FÃ‚Â£h
(2.66)
:= (/,tt(Vi))Ã‚Â£2 X (^K(g) + JF(iKnK)(f.^qi'\
KeTH ' K 'K
 X ^ Dr/ X'^(Vl))^
*6T ^
(2.67)
Following the symmetric general method presented in (2.50), we define a sym
metric version of method (2.65): Find (uupi) G Vi x Qd such that
Bl((i,Pi),(wi,9i)) = F 1{vuqi), (2.68)
for all (ui,gi) G Vi x Qd, where
B((ui,Pi),(ui,7i)) := B((7r(t*1),pi), (tt(i;i),
X rr2 (Nk(ul) ~ Pi + n^Pl),.^(*>l) <7i + ntf^))^
KÃ‚Â£Th atl*
 X ^(nf(IPil),nF(I9ll))F. (2.69)
As we know the exact expressions for ttk{ui) and Nk{u\), methods (2.65) and
(2.68) are fully discrete.
47
2.6.2 Methods requiring a twolevel solution
In this section, we assume the operator px{ui) is defined by problem (2.1) with
vF{oui) = aui. This yields the problem
Pk(ui) + VPtf(ui) = 0, Vpk(ui) = Viti in K (2.70)
Pk(ui) nF = tii nF on dK.
Notice the term {f,p(v\))K present in the righthand side of methods (2.46) and
(2.50). Recalling that pK(vi) = v\ + v, f = S7MpK(f), and problem (2.29)
(which defines vwe note the following:
(f,P/((v i))K = (f,v i+v)K
= (f,v1)K + (VMpK(f),v")K
= (/, v,)K (MpK(f),V v?)K + (MpK(f),v? nK)dK
= (f,v,)K. (2.71)
So, for this particular operator p( ), we may replace the term (/, p(v\)) on the right
hand side of (2.46) with the term (/, Vi).
We next observe that problem (2.70) does not have a closedform solution, and
so we must solve via a numerical method. Taking (px(ui), Vk(ui)) to be an approx
imate piecewise polynomial solution to problem (2.70), we see the general method
(2.46) (with the alternative righthand side) takes the form: Find (u1;pi) e Vj x Qd
such that
B2((u1,Pi), (ui,9i)) = F2(v1,71), (2.72)
48
for all (ui, <7i) G V] x Qd, where
B2((wi,pi),(ui,gi)) := B((p(ui),p1), (p^j),gi))
+ (*r(lbi]).^P/f(wi) + V<1i)k
KTh
~ J2 ('PKiui) Pi +UK{pl),VK(vl) qx HMg,))
KTh k
~ ^2 ^_(nf'(iPll)>nF([9lI))F,
FÃ‚Â£h F
F2(ui,<7i) := (/,i)n fn^(5) + ^2'(IFn^)(/x),g1N)
KTH \
1
(5^2/x.^(i)
FTh V
(2.73)
Again, following the symmetric general method presented in (2.50), we define a
symmetric version of method (2.72): Find (ui,pi) 6 V[ x Qd such that
B2((ui,Pi),(i>i,<7i)) = F 2{vuq1),
(2.74)
for all (v\,qi) 6 Vi x Qd, where
B2((ui,Pi),(vi,9i)) := B((p(u1),pi), (p(ui),
~ ^2 ~Jfp (Vk{ui) Pi + nK(pi),PAr(vi) q1 Hl/rfo
KCTh K
Ã‚Â£
FÃ‚Â£h
^^(nHIpiDiMIM))*,
In Chapter 4, we make the terms p/e(ui) and Vk{u\) explicit and provide an
analysis for this fully discrete method using an approximation approach presented in
[2]. It should be noted that as we solve for pk{u\) and Vk{ui) locally, methods
(2.72) and (2.74) are twolevel methods.
49
3. Error Analysis of the General Methods
We focus in this chapter on providing an analysis of the general methods (2.46)
and (2.50). Since these results apply for either choice of boundary condition up{u\
n) (See Subsection 2.1.1), this analysis will apply to any of the related methods in the
case that the expression for pk{u\) is known exactly. Therefore, methods (2.65) and
(2.68), for which Pk{u\) automatically inherit the analysis of this chap
ter. However, methods (2.72) and (2.74) involve approximations to pK{u\). This
approximation will introduce extra errors into their formulations which are not con
sidered in this analysis, and so this analysis does not apply directly to these methods.
These errors will be handled in Chapter 4.
The outline of this chapter is as follows:
In Section 3.1, we build upon standard interpolation results in order to bound
the interpolation error in the mesh dependent norm (2.52). Other important
bounds will be developed or cited for their use in later parts of the analysis.
Consistency results for both general methods are discussed in Section 3.2.
In Section 3.4, we present error estimates for both general methods presented
Section 2.4. Estimates are given in both the meshdependent norm (2.52) and
in natural norms. In the case a discontinuous pressure is used, the mass conser
vation properties of Section 3.3 are used in the proofs of these results for the
velocity.
50
In the following, we assume C, Cit C2 denote generic positive constants,
independent of H or cr, with values that may vary in each occurrence.
3.1 Interpolation results
Suppose u G [//(fl)]2, where 0 < t < 2. We will denote by V\ the Cldment
interpolant of u. The Clement interpolation operator (cf. [27, 22]) Ch : > V\
(with the obvious extension to vectorvalued functions), satisfies, for all K G TH and
all F G Ã‚Â£h.
IIq CH(q)IU,* < Cc2 HlK~m \q\tplK Vq G Hl{u>K), (3.1)
9 CH(q)\\o,F < Cc3 H'P \q\t,UF Vq G Hl{uF), (3.2)
and a stability condition,
C/ffa),,n < Ccl\\q\\ut Wq G H\Sl), (3.3)
for t 1,2, m 0, 1, where u>k {K' G TH : K n K' 0} and ojf = {K G 7w :
KnF^Q}.
Next, suppose q G Hl(fl) fl Lg(fi). In order to take into account the approxima
tion of the pressure, we consider the L2(Q) projection onto Qd which is denoted by
IIjj : L2(fl) > Qd. This projection satisfies (cf. [22])
lknS(g)m,n < Cp//mkMJ WqeHl(n), (3.4)
for 0 < m < t < 2. We will typically denote the interpolant by q\.
We wish to use these to find a bound on the interpolation errors in the mesh
dependent norm. For this, we will need a bound on the norm of the jump of the
difference between p and its interpolant q\.
51
Lemma 3.1 Letting q\ G Qd be an interpolant of p G 1 < t < 2, which
satisfies the approximation property (3.4), then there exists C > 0 such that
Proof: From the shaperegularity of the mesh, the definition of the jump, the trace
inequality, and (3.4), we can make the following bounds
and the result of the theorem is established taking the square root of both sides.
We will ultimately need a bound on ^2FÃ‚Â£h 7J7^lnF([pJ)llo,F which is present
in the meshdependent norm. Consider the following lemma, which is the first step
in establishing this.
Lemma 3.2 Given q G Lq(J1) fl H2(Th) there exist constants C5 and Ce such that
Proof: The first inequality follows directly from the stability of the projection onto
(3.5)
constants (see the appendix) using each edge F as a domain, taking C5 = . To
52
establish the righthand inequality, consider the following local auxiliary problems:
au + Vp = 0, V u = CK
U Tip
a
HF<7
[9] nF
in K,
on dK.
au + Vp = 0, V ii = Ck
ii tif = ^nF(M) nF
in K,
on dK,
where CK = Y^FCdK nF([[gJ) F is g'ven in Equation (2.22).
It may be established that
2 (Vu,9)k =
KTh
Likewise, we see
Ã‚Â£<*v)*+ E jrrlMlL
KTH FeeH F
E (Vu.?)K = E(*.v?)k+ E 7r7l(nfW)lS,F
KTh KTh FÃ‚Â£h nF
From this and the fact V u = V it, it is clear
53
We need a bound on JZk&th (w , Vq)/c in order to establish the result. To this end,
observe the Cauchy inequality, Lemma 2.12, mesh regularity, and Equation (3.4),
(u it, Vq)K = (V(p p), Vq)K
o
<\\^(pp)ioKm\o,K
So, using Equation (2.56) (with [[<71]] rather than Ilf ([[<71])) we get
E
K6Th F
Taking C6 = C%CTCP, we get the inequality we seek.
Lemma 3.3 Under the assumptions of Lemmas 3.1 and 3.2, and given q Hl(K),
it follows that
for 1 < t < 2.
FGÃ‚Â£h
1 1 1/2
J^mFdqqMl} < CH^\q\m,
(3.6)
Finally, we may establish the result on the interpolation error as measured in the
meshdependent norm. Using Corollary 3.3, we can establish an interpolation result
in the meshdependent norm.
54
Lemma 3.4 Suppose that (u,p) e ([Hl(Q)]2nH0(div, fi)) x (//2(f2)nLg(Q)) is the
solution to (1.12) and that (ui,<7i) Vi x Q\ are interpolants satisfying Equations
(3.1) and (3.4). Then, there exists C > 0 such that
l( ~ vuP ~ 9i)l < CH pl,n^ (37)
Proof: The result follows directly from the assumptions in Equations (3.1) and (3.4)
and from the result of Lemma 3.3.
In addition to these interpolation results, it will be useful to establish some results
regarding the operator Pk{) Since there are two such operators, we will note results
for each independently and then collect the properties which are shared.
Recalling that n(v) is used to denote the global RaviartThomas interpolation,
consider the classical interpolation estimate related to this operator [34, 22].
Lemma 3.5 Let K e 7//. There exists C > 0 such that
t> Mu)flo,K < CHk\v\uk,
for alive \H\K)f.
In addition to this, note the following lemma.
Lemma 3.6 Let K e Th and suppose Pk(v) has the property Pk{v)tif = vnF.
Then, for all v e [Hl{K)\2 there holds:
0 (pk(v),v)k = \pk{v)\1'K;
") \Pk{v)\I k = Ho,* \\v pK(v) fQ K;
iii) \aM%{v)\0tK < Ipk(v)\^k < w0i/f;
55
iv) \\v pK(v)\\0J< < ^\v\hK.
Proof: See [2],
The following corollary holds as a result of the previous two lemmas:
Corollary 3.7 There exist constants C[ and C% such that
*) \Pk(v)Io,k < ci\\v\o,K
u ~ Pk(v)\\0'K < C%Hk\v\uk;
for alive [Hl(K)}2.
Proof: The two results clearly hold due to Lemma 3.6 when we assume pK(v)nF =
v nF. Also, the two results hold by Lemma 3.5 by recalling pF(v) = nK(v) in the
case pF(v)nF = UF(vnF). m
3.2 Consistency results
We would like to establish results concerning the consistency of the methods. It
turns out that the bulk of the work can be done for the symmetric general method, and
the result for the general method will follow. We see that each of these methods is
consistent. First, consider the result for the symmetric general method (2.50).
Lemma 3.8 Let (u,p) e H0(div,Q) x //2(f2) n Lg(f2)] be the weak solution of
(1.12) and {u\,p\) be the solution from the symmetric general method (2.50). Then
B9s((u uup pi),(wi,r!)) = 0,
V(tOi,ri) G Vi x Qd.
56
Proof: Under the assumptions of the lemma, we see that [p]f = 0 on each internal
edge F and RM(u,p, /)a = 0 and Rc(u,g)\K = 0 for all K G 7//. Using Remarks
2.7 and 2.9, the facts RM(u,p, f) = 0 and Rc(u,g) 0, and definition (2.49) of
B, it follows,
Bf((u,p),(u7i,r1)) = = B((p(u),p), (p(wi),ri)) E H2^k{u) p + UK(p),VK(wl) ri + IItf(ri))K k&th a K  ^2 (v u n*(v u),vK(wi) n + nK(ri))jr KeT = B((u Mu{RM),p), {p{wi), ri))  E 4rT(Mp(RMirK(w1)r1+UK(r1))K KTh atl* + Y, (V RC nK{FF),VK{w{) n + UK(ri))K KerH  E (5 n*(s) n*)(/ X), Vk{wi) n)* = B((u,p),(p(tii),r1)) E^ nx(tf) W2 (J* ntf)(/x),7MM n)* /C6TW Ut1K = (f,p(wi))n E (n*(s) + rr2 (2if nK)(/x),r1)K *er KTH = Bf((t*i,pi), (twi.n)).
The result follows directly from this.
As a corollary to the last result, we may establish a consistency result for the
general method (2.46).
57
Lemma 3.9 Let (u,p) G H0(div,Q) x [H2(Q) n Lg(fi)] be the weak solution of
(1.12), (iii, pi) be the solution from the general method (2.46). Then
Bs((uu1,ppi),(i1,r1)) = 0,
V(wi,n) G Vi x Qd.
Proof: Under the assumptions of the lemma, we see that [p]f = 0 on each internal
edge F and RM(u,p, /)\K = 0 and Rc(u,g)\K = 0 for all K G TH. Therefore, by
the proof of Lemma 3.8 we get
B9((u,p),(iOi,ri)) = Bf((u,p),(io1,r1))
= (/, P(wi))n  + ^r(/ x + Ck) ri)k
K
KerH
 hrif x + ck)^v<(w1))
KTh
= B9((iii,pi),(ii7i,r1)).
K
The result follows directly from this.
Remark 3.10 Another form of consistency exists. We recall problem (2.24): Find
{uHiPh) G (Vi + Ve) X (Qd + Qe) such that
B((uH,pH),{wH,rH)) = F(wH,rH), (3.8)
for all Wh = p{w\) + wb G p(Vi) Vb and rH = + rb e Qd Wx. Notice by
the inclusions
p(V,) V6 C H0(div, n) and Qd 0 W C Lg(fi),
58
it follows that the exact solutions (u, p) to problem to problem (1.12) satisfy
B((u,p), (wH,rH)) = F{wh, rH),
from which it follows
B((uuH,pPH),(wH,rjf)) = 0, (3.9)
for all Wh = p(wx) + wb Ã‚Â£ p(V i)Vb and r h = ri + rb Ã‚Â£ QdWL. In particular,
we see
B((u uH,ppH),(p(wi),ri)) = 0, (3.10)
for all p(wi) Ã‚Â£ p(Vi) and for all r1 Ã‚Â£ Qd This result applies to (uh,Ph)
(t*i + wf + tif + uf ,pi + p^ + Pe + Pe1) formed using solutions (tii,pi) of the
general method (2.46).
Suppose that (iii, pl) solve the symmetric general method (2.50). This method is
derived by neglecting the term
X + Vrx)if.
KTh
So, we may write that (Uh,Ph) satisfies the modified form of (2.24): Find (u h,Ph) G
(Vi + Ve) x (Qd + Qe) such that
B((u//,Ph), (ti//,rj/)) ^ (u,op(wi) + Vr1)K = F(wH,rH), (3.11)
kgt
for all wH = p(wi) + wbÃ‚Â£ p(Vi) V* and rH = + rb Ã‚Â£ Qd W. Using the
fact that B((u,p),(p(wi),r1)) = F(p(wi),ri) for all (tui.n), we find:
B((ti Uh,P Ph), (p(wi),ri)) X (tifVp(>i) + VrQjc, (3.12)
KCJh
for all Wh p(wi) e p(Vi) and rtI = r\ Ã‚Â£ Qd. We conclude that the symmetric
general method does not share the consistency result (3.10) with the general method.
59
Remark 3.11 Notice the types of consistency results in Theorems 3.8 and 3.9 versus
the consistency results of Remark 3.10 are qualitatively different from one another.
The consistency results of the theorems indicate how the exact solutions with the
relevant assumptions act inside the bilinear forms in the definitions of the numerical
methods. This contrasts with the consistency results of the remark, which indicate
how the numerical solutions act in the bilinear form definining the weak form of the
Darcy problem.
3.3 Mass conservation property
Notice that methods (2.46) and (2.50) are designed to ensure that Uh Vi + Ve
satisfies
for all qi Qd. In the case that discontinuous pressure interpolation is used, we see
that this implies in particular that
for each K gTh, and we see the methods are mass conservative. We document this
in the following two lemmas:
Lemma 3.12 Let (ui,pi) be the solution to the general method (2.46) (respectively,
(3.56)). Then, supposing the discontinuous pressure case, it holds,
(V uH,qi) = {g,qi),
(3.13)
(3.14)
where uf is given by (2.37).
60
Proof: This result follows directly from Equation (3.14), and the two facts that
V u= 0 and JK V uf = 0 (see (2.29) and (2.30)).
Lemma 3.13 Let (ui,pi) be the solution to the symmetric general method (2.50)
(respectively, (3.58)). Then, supposing the discontinuous pressure case, it holds,
Remark 3.14 The above results are interesting. Although U\ is not itself locally mass
conservative, we may update it with in order to recover this feature. We shall see
that this property is useful in establishing convergence results as well.
3.4 Error estimates
In this section, we show the methods are optimally convergent. In what follows,
we define the errors:
eu:=up(ui), ep:=ppu
where (p(ui),pt) is the solution to the symmetric general method (2.50). We also
define the interpolation error as:
r]u:=uv i, T)p:=p~qu
where v\ and q\ are interpolants which satisfy Equations (3.1) and (3.4), respectively.
where u is given by (2.37).
Proof: Same as the proof of Lemma 3.12.
61
We first consider results for the symmetric general method (2.50) in Subsection
(3.4.1). We find in Theorems 3.15 and 3.19 that (p(ui),pi) and (uH,pH), respec
tively, converge in the meshdependent norm (2.52). Theorem 3.20 outlines the con
vergence properties of p\ and pn in the L2norm, and Theorem 3.26 shows the con
vergence of the locally mass conservative velocity of Section 3.3. As preparation to
prove the results of Theorems 3.19 and 3.20, some results are developed in Theorems
3.21, 3.22, and 3.23. These last theorems will be referred to in other sections as well.
Next, we present in Subsection 3.4.2 an analysis of the general method (2.46),
with convergence results in the meshdependent norm presented in Theorems 3.27
and 3.28. Results for convergence in natural norms is given in Theorems 3.29 and
Finally, we will show in Subsection 3.4.3 that one term on the righthand side
of the methods may be neglected in the general methods (2.46) and (2.50) due to its
small impact on the error.
3.4.1 Estimates for the symmetric general method
We begin by showing the convergence of (p(v.i),pi) in the meshdependent norm
Theorem 3.15 Let (u,p) ([Z/1 (Ã‚Â£1)]2 fl H0(div,Cl)) x (H2(Sl) x Ll(Sl)) be the
solution of (1.12) and (ui,pi) 6 V] x Qd be the solution of method (2.50). Then,
there exists C > 0, independent of H, a, and a such that
Proof: Let V\ and q\ be interpolants of u and p, respectively. We have the following
bound on the errors in the meshdependent norm, using its definition (2.52), the trian
3.31.
(2.52).
62
gle inequality, and the interpolation results of Lemma 3.4 and part (ii) of Corollary
3.7,
lll(eu>ep)lllw = ll(7? + t,i p(vi) + p(vi)p(ui)iVp + <1i Pi)Ih (315)
+t(Vu,Vp)t2H (3.16)
< C ((p(vi ui)i 7i Pi)111// + ffl(i p(i),0)a)
+C\\{nu,T]p)fH (3.17)
< Cl(p{vi Ui), qi pi)\fH + CH2a\u\\n
+CH2 J2 Hp,s2^ (3.18)
So, we see the bound on the error in the meshdependent norm is given up to the term
HI i),<7i Pi) III // We now seek a bound on this in terms of the interpolation
error. To shorten the notation, define W\ := V\ U\ and ri := q\ p\. Using the
stability result of Lemma 2.15, the consistency result of Lemma 3.8, and the definition
(2.49) of Bf, we have
/3(p(tui),ri)^
= Bf((ii! u,pi p), (toi, n)) + B((u vi,p gi), (toi, n))
< BJ((j,n,),(tDi,r1))
= l(Mto).P(i))n + (V 77, (V (u7i),77p)ji
 ^2 ^2(^(7?)/7p + nA:(7yp),7>Ar(tUi) + r1n,s:(ri))/(r
KTh a K
63
We will now treat each of these terms separately. First, Youngs inequality and the
stability of p() (part (i) of Corollary 3.7) imply
WVu),p(v>i))n < Y
< Cj0\\r)u\\ln + 2^ffp(wi)lSiii (319)
Next, integration by parts, identity (1.26), Youngs inequality, and Lemma 3.2 imply
(V7iu,ri)n = Y,b*,Vrl)K+ E IM)f
KTh
72 I, h2 , 1
FÃ‚Â£h
< ^ E iVrii, + Ã‚Â£ E
Ea
KTh
2 , 73
2a
FeÃ‚Â£H
FzÃ‚Â£h
2Hfo')z
2
0 ,F
< y^hllo,n + ^ E lVrilo
KTh
+ J^ E aH}< (l7^lo,/f + hk\vJiIk
4q \Hk
E lVrllo.K + E 7^;ln*(Irll)lo,.
+
Cq
273(7
KKTh FÃ‚Â£h
< I'ML + 2^ E lVr'lM
KTH
tfT
+
Cq
273(7
E Iloilo,K+ E ^lln^(^l)lo,Fj (320)
KK
FÃ‚Â£Ã‚Â£h
64
Next,
 (V"U71,77p)n= 5^(p(u>i),V r]p)K J^(/t(iOi),[n,l)F
 Y ^\pM\1,k+ Y Ã‚Â£iiv^Ho,
KTh 14 KTh
K
< E + E
(
47s
75
II Milo, F
E E
KT
FeÃ‚Â£H
(3.2D
Next, Youngs inequality, the triangle inequality, the fact that Vk{u 1) E Lq{K) to
gether with the Poincare inequality, problem (2.1) (pk(u\) = ^Wk{ui)), and the
stability result for p() imply,
 Y, i/2 (^Kivu) ~Vp + nK(f/p),P)f()i) + n ii/c(r 1 ))/c
KTH aMK
< E
KTh k
1
 Tip +
+
2aH2K^\Vk(w'} + r'~ n*(ri)l
E {^JFK{nMl,K + ^K^K(nX.K)
c
7 e
S^'Vr'Ã‚Â£
lo
\K
65
Finally,
FÃ‚Â£h F
F Ã‚Â£h
Taking each of these inequalities together, we have
\l,F
(3.23)
2^ +76)M0,n + 7 2 Hkl^li
KTh
.. i
+
T , . ,73
2(7i+72 + ^
,K
E ^t(7. + C76))IVr,X,K
KTh
+ E 27iT (*11^ +
f
Co { 1
lo,F
+
*
Oo ( 1 1 1 IV. I2
T \^ + ^ + 9^ + ^l Ip('u,1)IIo,
2 \7i 74 275 76 /
+ E Ca 11
,K
KÃ‚Â£TH
V IS c'T.. I
+c(; E 1**
J
KÃ‚Â£Th
i2
; E IV
KÃ‚Â£Th FÃ‚Â£h F
G,f
(3.24)
when taking 71 = 72 = 74 = 76 = 77 := 7s := and 73 := ^ when a > 1
jj when 0 < a < 1. Applying the interpolation result in Lemma 3.4 and
Equations (3.1) and (3.4) and collecting like terms, we find
!(/>(.), r.)i
and 73 :=
66
Finally, we are able to determine from Equations (3.18) and (3.25) there exists C > 0
such that,
Remark 3.16 Notice again, as was remarked after the wellposedness of the symmet
ric general method was established, that there is in fact no dependence on a in this
method. As such, an implementation of the symmetric general method may assume
Remark 3.17 Notice that although the above theorem ensures the convergence of the
solutions (p(ui),pi) as H is refined, this convergence has not been established in the
natural norms. Rather, we have only shown, in particular, that Vpi converges to Vp.
The convergence ofpi in the L2norm will be addressed momentarily.
Remark 3.18 The theorem shows that we have achieved an optimal convergence of
p{u\) to the true solution in the L2norm, based on the regularity assumption on the
true velocity u.
Convergence properties other than that established in the above theorem hold.
In particular, we may show that the solution (Uh,Ph) also converges to the exact
solution in (, )//. Furthermore, we may show that ph and pi convergerge at order
H2 in the L2norm. These facts are stated in the next two theorems. The proofs will
be given after some preliminary results have been established.
(3.26)
67
Theorem 3.19 Let (ui,pi) 6 V] x Qd be the solution of method (2.50) and assume
the conditions of Theorem 3.22. Taking
Uh = p(tii) + 1if + 11 Ph = Pl +Pe +Pe + Pe ,
there exists C > 0, independent of H, a, and o such that
l( ~Uh,P~ Ph)Ih < CH{ 1 + a)i + ^=p2ln^
Proof: This is proved on page 76, after some preliminary results have been estab
lished.
Theorem 3.20 Let (ui,p\) Vi x Qj be the solution of method (2.50) and assume
the conditions of Theorems 3.22 and 3.23. Taking
uH = p{ui) + uf + uGe, p =pi+p?+p"+pf.
Furthermore, assume that f is constant throughout the domain. Then there exists
C > 0 such that,
lb Pw0,n < CH2 ^1 + (1 + a)i + bkn) ,
lb PiIIo,n < ch2 (1 + Q)^ + IpI2)
Proof: This is proved on page 76, after some preliminary results have been estab
lished.
The following three results (Lemma 3.21 and Theorems 3.22 and 3.23) must be
established in preparation to prove the above two theorems. We first prove a lemma
regarding the enrichment velocities and pressures. These results show, in particular,
that certain quantities related to the enrichment will converge to zero.
68
Lemma 3.21 Suppose (u,p) e ([//1(fi)]2 D H0(div,Q)) x (H2(Q) n Ll(Q)) is the
solution of (1. 12) and (vi, qj) Vi x Qd are arbitrary. Several estimates regarding
the enrichment velocities and pressures may be proved.
1. Let (v*r, qÃ‚Â£!) be solutions to problem (2.29). Then
VgeMo
2. Let be the velocity solution to problem (2.30). Then
\\vGi <
KIIo.kS ^
\u\i,K + 77 ( l PfaOlo.K + IV(P 9l)l0,K
Hk \ o
3. Let be the velocity solution to problem (2.31). Then
/ llUe llo,K
L kcth
1
2
<
* L FÃ‚Â£Ã‚Â£h t
1
2
4. Given qe = + q^ + q^{, there exists C > 0 such that
.KTh
< C [(1 + a)l\{u p(ti),p qi)\\l + H2o\u\ln] 5 .
Proof: We begin with the first result. By definition, the fact that
1 1 ,
u HVp = /,
a o
and the triangle inequality,
llVgfl =l~v^ + fvlVql\
= II t>" + U Vi + V(p 7i)
o 0 ,K
= lp(wi) + V(p9i)
(7 0,K
< u p(ti)0iK + ^V(p qi)\\o,K (327)
69
We now consider the second result. By problem (2.30), integration by parts, the
CauchySchwarz inequality, the Poincare inequality using <7^, qf 6 L\(K), and the
triangle inequality, it follows,
= (>)*
1
= \(fe,VVGe)K
<^ll?eGUIV^IU
1 Gil 11 nC/., rr y
K Q.K
(3.28)
<Il9e Io,kIr {vug)UK(R {vug)) ^q,
< C^ v,f 0iK  nK(flc(1,9))0K + ^rlrf'l,K)
< k (\Hctv,,g) n*(flc(t>.,))lwt + ^V
< CHKtv'Jl,K (v U imv u)0K + ^10Jf)
< CHkIvXk (V u0, + ^!V?"llc,,K) ,
and so we conclude from the first result and the definition of lull *,
IloilotK
7T [ hk \ 'a / J
which is the bound in the statement of the lemma.
Next, consider the third result. Using problem (2.31), Corollary 2.13, and the
proof of Lemma 3.1, we find,
E i?il
.KÃ‚Â£Th
K
1
E Ik
so^[E^i>wh
L FÃ‚Â£h
(3.30)
70
So, the third result is established by noting p] = 0 under the assumptions of the
theorem.
Finally, we prove the final result. Using Lemma 2.13, the above two results, and
problem (2.30), we have
Ã‚Â£ iv*;U= Ã‚Â£ ivgf + v, + v,flK
KTH
KTh
KTh
FÃ‚Â£Ã‚Â£h
+C(W p(vi),P <7i)//
= c(l + o)l( p(vi),p qi)fH + CH2cr\u\\iV (3.31)
The following theorem establishes the error committed by (vu,qn) is on the
order of the error committed by (p(vx), q\) in the meshdependent norm.
Theorem 3.22 Let (u,p) ([tf1^)]2 n H0(div, 

Full Text 
PAGE 1
RESIDUAL LOCAL PROJECTION METHODS FOR THE DARCY PROBLEM by Christopher E. Harder B.S., Metropolitan State College of Denver, 2003 M.S., University of Colorado Denver, 2006 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 2010
PAGE 2
This thesis for the Doctor of Philosophy degree by Christopher E. Harder has been approved by P. Franca Jan Mandel Date
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Harder, Christopher E. (Ph.D., Applied Mathematics) Residual Local Projection Methods For the Darcy Problem Thesis directed by Prof. Leopoldo P. Franca ABSTRACT When solving the Darcy problem in its mixed form, it is wellknown that the set of pairs of spaces chosen to approximate the velocity and vector fields is restricted by the necessary and sufficient LBB condition. In order to circumvent this restriction, stabilized finite element methods have been developed. In the tradition of the PGEM methods used to stabilize the IP\/JP>0 element for the Darcy problem, the Residual Local Projection (RELP) methods are developed as an approach to stabilization when approximating with equalorder linear spaces using both continuous and discontinuous pressure interpolations. A total of four related methods are presented, two of them being of the twolevel type. The methods are analyzed and shown to be optimally convergent. Furthermore, the methods employing a discontinuous pressure present a massconservative velocity. Finally, numerical tests are performed to validate the theory and show performance on a realworld problem. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed LeOpoldo P. Franca
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DEDICATION This dissertation is dedicated to members of my family, whom I have always been able to count on for support and encouragement through my many years of study.
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ACKNOWLEDGMENT Many thanks are owed to Dr. Leopoldo P. Franca for the opportunity to travel and meet with some of the top researchers in the field. Without question, this results from the fact that he resides among those at the top and I am very grateful to have had a chance to work with him. A debt of gratitude I could never hope to repay is owed to Dr. Frederic Valentin for his diligent and exceptionally patient oversight and contribution to the develop ment of this work. In the course of my studies here, I have met many wonderful people who have enriched my life professionally and personally. In particular, thanks to Dr. Bednch Sousedik and Minjeong Kim for the many nights of studying and discussions (and beer!). Thanks also to the other students, too numerous to list, with whom I have shared many nice experiences and whom I now count among my friends. I am also grateful to the department and its professors for the chance to gain insight and knowl edge from the interesting classes I have attended over the years. I would finally like to thank my friends and family for their patience during my years of academic distraction. In particular, thanks to my mother, sister, and grand parents for their longstanding support, my father for his curiosity, Diane for pushing me when things were tough, and my friends in general for offering a space to be relaxed. This would have been impossible without all of you!
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Figures. Tables Chapter 1. Introduction 1.1 Preliminary notation 1.2 The Problem . . 1.2.1 The Darcy Problem 1.2.2 Simplifying assumptions CONTENTS 1.2.3 Weak form of Darcy's problem 1.3 Discretization and approximation 1.3.1 Notation related to discretization 1.3.2 Numerical approximations 2. The RELP methods 2.1 Preliminaries . 2.1.1 2.1.2 A few operators A few subspaces 2.2 The enriching approach: Motivation 2.3 The enriching approach: Defining appropriate subspaces 2.4 The enriching approach: Derivation of the methods 2.4.1 Local problems in the method . . . . . vi X XVII 7 8 9 10 11 13 13 16 21 22 22 23 25 29 31 32
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2.4.2 Solutions to the local problems 2.4.3 Two general methods . . . 2.5 Existence and Uniqueness Results 2.5.1 Some auxiliary results . . 35 38 40 40 2.5.2 Existence of a unique solution for the symmetric general method 43 2.5.3 Existence of a unique solution for the general method 44 2.6 Explicit methods formed from the general methods 46 2.6.1 Methods with analyticallydefined terms 46 2.6.2 Methods requiring a twolevel solution 48 3. Error Analysis of the General Methods 50 3.1 Interpolation results 51 3.2 Consistency results 3.3 Mass conservation property 3.4 Error estimates ...... 3.4.1 Estimates for the symmetric general method 3.4.2 Estimates for the general method 3.4.3 Results for a general g E H1{D) .. 3.5 Concluding remarks . . . . 4. Error Analysis of the Fully Discrete Methods 4.1 The Fully Discrete Methods ....... 4.2 Numerical analysis of the fully discrete methods. 4.2.1 A relationship between two velocities. 4.2.2 Approximate enrichment solutions 4.2.3 Wellposedness results . . . vii 56 60 61 62 79 82 85 86 87 91 91 94 98
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4.2.3.1 Wellposedness of the fully discrete symmetric method. 4.2.3.2 Wellposedness of the fully discrete full method 4.2.4 Consistency results Convergence results. 4.2.5 4.2.5.1 4.2.5.2 Convergence results for the symmetric fully discrete method. Convergence results for the fully discrete full method. 5. Numerical Experiments 5.1 An analytical problem 5.1.1 The symmetric methods 5.1.1.1 Methods with an analytic M K 5.1.1.2 Methods with a nonanalytic MK 5.1.2 The full methods . . . . 5.1.2.1 Methods with an analytic M K 5.1.2.2 Methods with a nonanalytic MK 5.2 A second analytical problem 5.2.1 The symmetric methods 5.2.1.1 Methods with an analytic MK 5.2.1.2 Methods with a nonanalytic MK 5.2.2 The full methods . . . . 5.2.2.1 Methods with an analytic MK 5.2.2.2 Results with a nonanalytic MK 5.3 The quarter 5spot problem, constant permeability 5.3.1 The symmetric methods . . 5.3.1.1 Methods with an analytic M K. viii 99 100 102 108 108 118 122 124 125 125 131 141 142 144 147 148 148 151 159 159 161 163 164 164
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5.3.1.2 Methods with a nonanalytic MK 5.3.2 The full methods . . . . 5.3.2.1 Methods with an analytic MK 5.3.2.2 Methods with a nonanalytic MK 5.4 The quarter 5spot problem, checkerboard permeability 5.4.1 The symmetric methods . . 5.4.1.1 Methods with an analytic MK. 5.4.1.2 Methods with a nonanalytic MK .. 5.4.2 The full methods ........ 5.4.2.1 Methods with an analytic MK 5.4.2.2 Methods with a nonanalytic MK 6. Conclusion Appendix A. DEFINITIONS AND THEOREMS A. I Useful Definitions ..... A.2 Various Useful Inequalities A.3 Some notes on an enrichment velocity. B. SOME THEORY FOR OPERATOR PROBLEMS B. I Theory for symmetric problems .. 8.2 Theory for nonsymmetric problems References . . . . . . . . ix 169 174 174 176 178 179 180 182 183 183 186 187 190 190 190 192 196 196 199 206
PAGE 10
FIGURES Figure 1.1 A saturated porous medium with pressure held higher on the left than the right so that flow is directed toward the right. 2 1.2 The normal vector. . . . . . . . 14 1.3 The lefthand plot shows how the singular values are distributed, with the xaxis showing the values (no meaning is assigned to the yaxis). The righthand plot shows the value of the singular values along the yaxis. In this case, the xaxis indicates the position of the singular values in a vector ordering them from largest to smallest, with a total of I 023. . 17 1.4 A sample crisscross mesh and a curve showing better than CJ( H2 ) con vergence for PH in the L2norm on a family of this type of mesh. . . 20 5.1 The mesh used when using a refinement at the second level with linear interpolation. . . . . . . . . . . . . . . . . 123 5.2 A typical structured (left, H = }2 ) and unstructured (right, H il4) mesh. 125 5.3 Convergence plots using continuous pressure interpolation for problem I (structured mesh). . . . . . . . . . . . . . . 126 5.4 Contour plots of the absolute value of velocity u1 (left figure) and pres sure p1 (right figure) using continuous pressure interpolation for problem I (structured mesh). . . . . . . . . . . . . . . 127 5.5 Convergence plots using for the symmetric methods with analytic MK and continuous pressure interpolation for problem I (unstructured mesh). 128 X
PAGE 11
5.6 Convergence plots using discontinuous pressure interpolation for problem I (structured mesh). . . . . . . . . . . . . 129 5.7 Convergence plots for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh). 130 5.8 Convergence plots using continuous pressure interpolation for problem I (structured mesh, quadratic second level) .................. 134 5.9 Convergence plots using continuous pressure interpolation for problem I (unstructured mesh, quadratic second level). . . . . . . . 135 5.10 Convergence plots using discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). . . . . . . . 139 5.11 Convergence plots using discontinuous pressure interpolation for problem I (structured mesh, quadratic second level) ............... 140 5.12 Contour plots of the absolute value of velocity u1 (left figure) and pressure p1 (right figure) using continuous pressure interpolation for problem 2 (structured mesh) .............................. 149 5.13 Contour plots of absolute value of velocity u1 and rr( u1 ) for the method with analytic MK and continuous pressure interpolation (structured mesh).I65 5.14 Elevation and contour (with velocity vector field) plots of pressure p1 + for the method with analytic MK and continuous pressure interpolation (structured mesh). . . . . . . . . . . . . .. 166 5.15 Contour plots of absolute value of velocity u1 and rr( u1 ) for the method with analytic MK and continuous pressure interpolation (unstructured mesh) ..................................... 166 XI
PAGE 12
5.16 Elevation and contour (with velocity vector field) plots of pressure PI + for the method with analytic MK and continuous pressure interpolation (unstructured mesh) ........................... 167 5.17 Contour plots of absolute value of velocity ui and 1r( ui) for the method with analytic MK and discontinuous pressure interpolation (structured mesh). . . . . . . . . . . . . . . . . . 167 5.18 Elevation and contour (with velocity vector field) plots of pressure p1 + + pf for the method with analytic M K and discontinuous pressure interpolation (structured mesh). ..................... 168 5.19 Contour plots of absolute value of velocity ui and 1r( ui) + uf for the method with analytic MK and discontinuous pressure interpolation (un structured mesh). . . . . . . . . . . . . . . 168 5.20 Elevation and contour (with velocity vector field) plots of pressure PI + + pf for the method with analytic MK and discontinuous pressure interpolation (unstructured mesh). .................... 169 5.21 Contour plots of absolute value of velocity ui and un for the method with analytic MK and continuous pressure interpolation (structured mesh).l70 5.22 Elevation and contour (with velocity vector field) plots of pressure PH for the method with analytic MK and continuous pressure interpolation (structured mesh). . . . . . . . . . . . . . . . 170 5.23 Contour plots of absolute value of velocity ui and un for the method with nonanalytic MK and continuous pressure interpolation (unstructured mesh). . . . . . . . . . . . . . . . . 171 xii
PAGE 13
5.24 Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and continuous pressure interpolation (unstructured mesh). . . . . . . . . . . . . . . 171 5.25 Contour plots of absolute value of velocity u 1 and uH for the method with nonanalytic MK and discontinuous pressure interpolation (structured mesh). . . . . . . . . . . . . . . . . 172 5.26 Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and discontinuous pressure interpolation (structured mesh). . . . . ................. 172 5.27 Contour plots of absolute value of velocity u 1 and uH for the method with nonanalytic MK and discontinuous pressure interpolation (un structured mesh). . . . . . . . . . . . . . . 173 5.28 Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and discontinuous pressure inter polation (unstructured mesh). . . . . . . . . . . . . 173 5.29 Contour plots of absolute value of velocity u 1 and 7r(u1 ) + uf for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation (structured mesh). . . . . . . . . . . 174 5.30 Elevation and contour (with velocity vector field) plots of pressure p1 + + pf for the full method (a= 0.1) with analytic MK and discontinuous pressure interpolation (structured mesh). . . .......... 175 5.31 Contour plots of absolute value of velocity u 1 and 7r( u1 ) + uf for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation (unstructured mesh). .................... 175 Xlll
PAGE 14
5.32 Elevation and contour (with velocity vector field) plots of pressure PI + + pf for the full method (a= 0.1) with analytic MK and discontinuous pressure interpolation (unstructured mesh). 5.33 Contour plots of absolute value of velocity ui and u1 + + uf for the 176 full method (a= 0.1) with nonanalytic MK (structured mesh, quadratic second level). . . . . . . . . . . . . . . . 176 5.34 Elevation and contour (with velocity vector field) plots of pressure PI + +pf for the full method (a= 0.1) with nonanalytic MK (structured mesh, quadratic second level). . . . . . . . . . . . 177 5.35 Contour plots of absolute value of velocity ui and u I + + uf for the method with nonanalytic MK and discontinuous pressure interpolation (unstructured mesh, quadratic second level) ................. 177 5.36 Elevation and contour (with velocity vector field) plots of pressure PI + + pf for the method with nonanalytic MK and discontinuous pressure interpolation (unstructured mesh, quadratic second level). ..... 178 5.37 Regions in the checkerboard problem. a = 1 in zones II and IV, and a = 0.001 in zones I and III. . . . . . . . . . . . . 179 5.38 Elevation plots of pressures PI and p1 +pf for the method with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . . . . . . . . . . . . 180 5.39 Contour plot (with vector field of the corresponding velocity) of pres sures PI and PI+ + pf for the method with analytic MK and discon tinuous pressure interpolation for the checkerboard problem (structured mesh) ..................................... 181 XIV
PAGE 15
5.40 Elevation plots of the pointwise Euclidean nonn ofu1 and u1 for the method with analytic M K and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . . . . . 181 5.41 Crosssections of the pointwise Euclidean nonn of u1 and u1 + + at x2 = .25 for the symmetric method with analytic M K and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 182 5.42 Elevation plots of pressures p1 and p1 + + for the method with nonanalytic MK (quadratic second level) and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . . . 182 5.43 Contour plot (with vector field of the corresponding velocity) of pressures p1 and p1 + + for the method with nonanalytic M K (quadratic second level) and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . . . . . . . . 183 5.44 Elevation plots of pressures p1 and p1 + + for the method full method (a = 0.1) with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . ....... 184 5.45 Contour plot (with vector field of the corresponding velocity) of pres sures p1 and p1 + + for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). . . . . . . . . . . . . . 184 5.46 Elevation plots ofthe pointwise Euclidean nonn of u1 and u1 for the full method (a = 0.1) with analytic MK for the checkerboard problem (structured mesh). . . . . . . . . . . . ... 185 XV
PAGE 16
5.47 Crosssections of the pointwise Euclidean norm of u1 and u1 + + at x2 = .25 for the full method (a= 0.1) with analytic MK and discon tinuous pressure interpolation for the checkerboard problem (structured mesh) ..................................... 185 5.48 Elevation plots of pressures p1 and p1 + + for the full method with nonanalytic MK (a = 0.1) for the checkerboard problem (structured mesh, quadratic second level). . . . . . . . . . . . 186 5.49 Contour plot (with vector field of the corresponding velocity) of pres sures p1 and p1 + + for the full method with nonanalytic MK (a= 0.1) for the checkerboard problem (structured mesh, quadratic second level). . . . . . . . . . . . . . . . . . 186 xvi
PAGE 17
TABLES Table 5.1 Table of convergence results for the symmetric method with analytic M K and continuous pressure interpolation for problem I (structured mesh). 127 5.2 Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem I (unstructured mesh) ..................................... 128 5.3 Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 1 (structured mesh). . . . . . . . . . . . . . . . . . 129 5.4 Local mass conservation for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 1 (structured mesh). 130 5.5 Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 1 (unstructured mesh). . . . . . . . . . . . . . . . . 131 5.6 Local mass conservation for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh) ..................................... 131 5.7 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 1 (structured mesh, linear second level). . . . . . . . . . . . . 132 xvii
PAGE 18
5.8 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (unstructured mesh, linear second level). . . . . . . . . . . . . 133 5.9 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (structured mesh, linear second level w/ refinement). . . . . . . . .. 133 5.10 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (unstructured mesh, linear second level w/ refinement). ................ 134 5.11 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (structured mesh, quadratic second level). . . . . . . . . . . . 135 5.12 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (unstructured mesh, quadratic second level). . . . . . . . . . . . 136 5.13 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (structured mesh, linear second level). . . . . . . . . . . ..... 136 5.14 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (unstructured mesh, linear second level). . . . . . . . . . . . 137 5.15 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (structured mesh, linear second level w/ refinement). ................ 138 XVIII
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5.16 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (unstructured mesh, linear second level w/ refinement). . . . ....... 139 5.17 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 1 (structured mesh, quadratic second level). . . . . . . . . . . . 140 5.18 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 1 (unstructured mesh, quadratic second level). . . . . . . . . . 141 5.19 Table of convergence results for the full method (a = 0.1) with an analytic MK and discontinuous pressure interpolation for problem I (structured mesh). . . . . . . . . . . . . . . . . 142 5.20 Performance of the full method with analytic M K and discontinuous pressure interpolation for problem I with different values of a (structured mesh). . . . . . . . . . . . . . . . . . 142 5.21 Local mass conservation for the full method with an analytic MK (a= 0.1) and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). . . . . . . . . . . . 143 5.22 Table of convergence results for the full method (a = 0.1) with an analytic MK and discontinuous pressure interpolation for problem I (un structured mesh). . . . . . . . . . . . . . . . 143 5.23 Performance of the full method with analytic MK and discontinuous pressure interpolation for problem I with different values of a (unstructured mesh). . . . . . . . . . . . . . . . . 144 XIX
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5.24 Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (unstructured mesh) ..................................... 144 5.25 Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). . . . . . . . . 145 5.26 Perfonnance of the full method with nonanalytic MK and discontinuous pressure interpolation for problem I with different values of a (structured mesh). . . . . . . . . . . . . . . . . 145 5.27 Local mass conservation for the full method with nonanalytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). ...................... 146 5.28 Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh, quadratic second level) ................... 146 5.29 Perfonnance of the full method with nonanalytic MK and discontinuous pressure interpolation for problem I with different values of a (un structured mesh). . . . . . . . . . . . . . . 147 5.30 Local mass conservation for the full method with nonanalytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (unstructured mesh, quadratic second level). ...................... 147 5.31 Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem 2 (structured mesh).148 XX
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5.32 Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem 2 (unstructured mesh) ..................................... 149 5.33 Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 2 (structured mesh) ..................................... 150 5.34 Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh) .................................. 151 5.35 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, linear second level). . . . . . . . . . . . . 151 5.36 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, linear second level). ........................ 152 5.37 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, linear second level w/ refinement). . . . . . . . . 152 5.38 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, linear second level w/ refinement). ................ 153 5.39 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, quadratic second level). . . . . . . . . . . . 153 XXI
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5.40 Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, quadratic second level). . . . . . . . . . . . 154 5.41 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, linear second level). ....................... 154 5.42 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, linear second level) ....................... 155 5.43 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, linear second level w/ refinement). . . . . . . . . 156 5.44 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, linear second level w/ refinement). . . . . . ... 157 5.45 Table of convergence results for the symmetric method with nonanalytic M K and discontinuous pressure interpolation for problem 2 (structured mesh, quadratic second level). . . . . . . . . . . . 158 5.46 Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, quadratic second level). . . . . . . ....... 159 5.47 Table of convergence results for the full method (a: = 0.1) with an analytic M K and discontinuous pressure interpolation for problem 2 (structured mesh) .................................. 160 XXII
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5.48 Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem 2 (structured mesh) ..................................... 160 5.49 Table of convergence results for the full method (a = 0.1) with an analytic MK and discontinuous pressure interpolation for problem 2 (un structured mesh). . . . . . . . . . . . . . . 161 5.50 Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem 2 (unstructured mesh) ..................................... 161 5.51 Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, quadratic second level). . . . . . . . . 162 5.52 Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem 2 (un structured mesh, quadratic second level). . . . . . . . . . 163 5.53 Local mass conservation for the symmetric methods using discontinuous pressure interpolation for the quarter 5spot problem with constant permeability. . . . . . . . . . . . . . . . . 164 5.54 Local mass conservation for the full methods (a = 0.1) for the checkerboard quarter 5spot problem. . . . . . . . . ....... 174 5.55 Local mass conservation for the symmetric methods using discontinuous pressure interpolation for the checkerboard quarter 5spot problem. 180 5.56 Local mass conservation for the full methods (a = 0.1) for the checkerboard quarter 5spot problem. . . . . . . . . . . . 183 xxiii
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1. Introduction The study of the flow of a fluid through a porous medium is of particular interest in many fields of research. In the study of ground water contamination plumes or the extraction of oil, the porous medium is the ground beneath our feet. This ground may be composed of rock, sand, soil, clay, or any combination of these, and is far from a solid, impermeable mass. Instead, gaps exists throughout, making space for oil or aquifers, etc. Naturally, the fluid in question needs to move through this medium in some way, leading us to question how best to model this. Furthermore, once a model is in hand, how do we get solutions from the model which are meaningful? Suppose we are given a porous medium which is completely saturated with a fluid (e.g., water, oil, etc.) which is flowing through it (see Figure 1.1 ). One model for describing such flows is attributed to Darcy, and in the case of a fluid in lami nar flow, this seems to be a good model. In its primal form, this model, also known as the Darcy problem, consists of a Poisson equation for pressure. Although this problem may be effectively solved using standard Galerkin techniques, it is often the flow velocity, which is related to the gradient of the pressure, is typically the quantity of interest. For instance, a common model of the miscible transport of an underground contaminant in water is a nonlinear system composed of the Darcy prob lem and a diffusionconvectionreaction equation with coefficients depending on the locally conservative flow velocity (from the Darcy portion). Furthermore, the numer ical approach should be mass conserving in order to be physically consistent, again requiring an accurate velocity profile. Therefore, methods which solve for the veloc
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6 6 Figure 1.1: A saturated porous medium with pressure held higher on the left than the right so that flow is directed toward the right. ity directly, as opposed to obtaining it through differentiation, are of intereset. This leads us to consider solving the Darcy problem in its mixed fonn. Approximation for the mixed fonn is not without restrictions. When approaching the solution numerically, one must ensure the matrix equation has a unique solution, a property which is not directly inherited from the continuous problem. The basic requirement is the LBB condition (or infsup condition) [14], which describes therelationship that must exist between the pressure and velocity approximation spaces. In addition to issues related to invertibility, the value of using higherorder polynomials is questionable when the exact solution doesn't have very nice regularity properties (a typical property of all finite element methods). In fact, loworder, stable elements which are locally mass conservative exist in a delicate balance, and many intuitive choices simply fail to satisfy all three properties. In order to satisfy the LBB condition, yet still use loworder approximation spaces, one may consider using loworder RaviartThomas (RT) or BrezziDouglasMarini (BDM) elements for discretization of the velocity. (See [34] and [ 15], respec2
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tively. See also [12, 22, 20] for a nice discussion, as well as [4] for an implementa tion of loworder RT elements in Matlab.) These elements are constructed to yield velocities with continuous normal components and (possibly) discontinuous tangen tial components across mesh faces, and have the advantage of being locally mass conservative and LBBstable when combined with pressure spaces of discontinuous polynomials of order one less than the velocity space. However, the approach leads to systems which require extra effort to solve efficiently [ 17]. Other loworder elements are of interest. Unfortunately, many of the most intu itive choices simply fail to satisfy the LBBcondition. The nodally defined 1P't/IP'0 IP't/IP'I. and IP't/JP"fisc (the last element indicating linear, continuous velocity together with linear, discontinuous pressure, respectively) are not stable pairs. However, inter est in them persists because they are: simple to program and they exist in standard codes; loworder, thereby keeping the number of degrees of freedom, and therefore the overall cost of computation, down. Many approaches consider stabilizing such pairs (among other pairings). One possibility is the consistently stabilized methods of [31, 29] for the Darcy problem using any combination of standard, nodallydefined elements for the velocity and pressure approximations. The approach is quite elegant and the stability coefficient is simply equal to 4. (See [ 18] for a more thorough analysis). However, the approach does not give a locally mass conservative velocity when continuous velocities are considered. 3
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Contributions from the area of leastsquares finite element methods exist as well. Inspired by work stabilizing the Stokes problem (see [21, I 0]), elements not satisfying the LBB condition were stabilized for the mixed fonn of the Darcy problem in [II] through the use of local projections of the pressure polynomial solution of order k into the discontinuous space of polynomials of order k1. In fact, this type of projection will be one of the key elements of the stabilization presented in this work. None of the above considered the use of the simplest, although unstable, IP\/1?0 element. A stabilization for this element has been considered in [5, 6, 2] in the frame work of PetrovGalerkin Enriched methods (PGEM). PGEM has been used to stabi lize discretizations of several operators (see, e.g., [25, 7, 5]). Motivated by the notion that instabilities in numerical solutions arise from an inability of the mesh to resolve fine scales present in the true solution, the underlying philosophy is to decompose the solution space into a polynomial portion and a fine(or sub) scale portion, as done in the Variational Multiscale Method (VMS) [28] and ResidualFree Bubble Method (RFB) (see, e.g., [ 19]). Through a static condensation procedure, the fine scale solu tions are then defined in tenns of local residuals of the operator. The distinguishing characteristic for PGEM is its stabilization via the incorporation of edge residuals into the fine scale solutions, achieved by choosing the boundary conditions for the local problem in tenns of these edge residuals. When applied to the reactiondiffusion op erator, the imposition of such nonhomogeneous boundary conditions stands in con trast to the RFB method, which requires the local solutions to satisfy homogeneous boundary conditions. Although very successful, RFB's still presented nonphysical oscillations in the solution in some numerical examples which are thought to be the result of the nonphysical boundary condition. The PGEM for this operator may be 4
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seen to be as a generalization of the RFB approach which seeks to replace the homo geneous boundary conditions with more physical ones. The resulting methods have shown an improvement in numerical tests (see [25, 24]). Returning to the stable PGEM methods based upon the IP\/IP0 element used in the mixed Darcy problem, we see the methods have many appealing characteristics: 1. They are derived via an enrichment of the standard approximation spaces. 2. They use loworder, nodallydefined elements readily available in finite element codes. 3. Symmetric methods are available in [6, 2]. 4. The continuity of the velocity is relaxed to simply the continuity across edges in the normal direction thanks to one portion of the enrichment solutions. 5. The derivation indicates a locally massconservative velocity. 6. Optimal convergence is achieved, and "superconvergence" is observed in some numerical examples. Based upon the success with such methods derived in the PGEM approach, fur ther exploration was considered warranted, and the ideas have been extended to the Stokes operator using, in particular, the IP\/IP\ and IPJ/JP1isc elements [8]. The present work considers a new extension of the PGEM to the unstable IPJ/IP1 and IPJ/JP1isc el ements used in the Darcy problem. The methods considered will be derived in the PGEM framework, and are called Residual Local Projection (RELP) methods. They may in some sense be viewed as a combination of ideas in [31, 18] (consistent sta bilization using the residual of the momentum equation), [ 11] (use of the projection 5
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into the piecewise constants), and importantly [5, 2, 6] (use of edge residuals to drive enrichment solutions), though the stabilizing tenns are ultimately different in fonn. In philosophy, the new methods can be seen as a generalization of the ideas present for the PGEM methods for the IP'I/IP'0 element, but significant differences appear as well: the linear interpolation of pressure chosen here requires extra care in order to control the gradient of pressure in the formulation; tenns involving the jumps have a different coefficient than is present in the PGEM for IP'I/IP'o. The outline of this work is as follows: After some introductory material, we de rive two general RELP methods in Sections 2.22.4 using an enrichment approach. The methods are general in the sense that a total of four (related) methods arise from them: two of the methods have enrichments which are analytically defined and so lead to methods which are fully discretized (see (2.65) and (2.68)); the other two require an extra approximation locally and so are derived as twolevel methods (see (2.72) and (2.74), or (4.17) and (4.14)). Under the assumption no approximation at the second level is required (i.e., all tenns in the method are known exactly), the com mon framework used in the derivation of the four methods is employed to show the methods have unique solutions (see Lemmas 2.15 and 2.17), as well as to provide an analysis for all methods. This analysis shows the methods to be optimally convergent (see Theorems 3.15, 3.20, 3.27, 3.29). In addition to this analysis, the analysis for the twolevel methods is considered, and all convergence properties are inherited for this version (see Theorems 4.13, 4.16, 4.18, and 4.19). It is interesting to note that 6
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the methods employing the IP\/JP1iBc element yield a locally mass conservative ve locity, an important feature used when showing the convergence of a velocity in the H(div, n) norm (see Theorems 3.26, 3.31, 4.17, and 4.21 ). Finally, we will consider numerical tests which validate the theory in Chapter 5. We also explore performance on the 5spot problem in the case of both a homogeneous and a heterogeneous porous medium. The rest of this introductory chapter is devoted to: I) describing the Darcy model in a general, and then more simplified form, as well as recalling the results used to establish the existence of a unique solution, 2) introducing notation for the Darcy problem and for discretization, 3) and closing with a few numerical methods for solv ing the mixed form of the Darcy problem. 1.1 Preliminary notation AssumeD is an arbitrary bounded set in !Rd. We will denote by L2(D) the usual space of square integrable functions over D. For functions v, w E L2(D), we define the inner product ( ) v and II II D by (v, w)v := l vw, llull; := (v, v)v. (1.1) We shall use to denote functions belonging to L2(D) with zero average in D (i.e., fv q = 0 if q E We also have a need for the space [L2(D)j2 L2(D) x L2(D) with inner product and norm defined by (v, w)v := l v w, := (v, v)v. (1.2) Given multiindex a, the inner products for the spaces H8(D), which contains func tions in U(D) with all derivatives of order 0 < Ia I :::; sin L2(D), will be denoted by 7
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(, )s,D and the induced nonns by II lls,D For v, wE H8(D), we take these to have definitions (v, w)s,D = L (ff'v, ff'w)o, llvii;,D := L (ff'v, ff'v)o ial$s ial$s Furthennore, we define the space H(div, D) := { v E [L2(D)J2 : '\1 v E L2(D)}, and we define the associated inner product and the nonn by (v, w)div,D := (v, w)o + ('V v, 'V w)o, := + II'V The space H0(div, D) stands for the space of functions belonging to H(div, D) which have nonnal component vanishing on 8D. We also recall the diameter of Dis defined by diamD = sup{lxYl: x, y ED}, where I I is the Euclidean nonn in JRd. Unless otherwise specified, 0 denotes an open bounded domain in JR2 with polyg onal boundary 80, and X = (xi, X2) is a typical point in f2. 1.2 The Problem The goal of this work is to develop stabilized methods for the Darcy problem and numerically test these methods both for consistency with the theoretical results to be proved herein, and for perfonnance on a realworld problem. In this section we introduce the problem ( 1.7),( 1.8). We then cast Darcy's problem in a weak fonn ( 1.12) and show that this fonnulation is wellposed in Theorem 1.12. 1.2.1 The Darcy Problem 8
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The Darcy problem is a set of equations relating a fluid's velocity to pressure, gravitational, and source terms. First, Darcy's law describes the relationship between the filtration velocity u (we simply refer to this as the velocity) and the gradient of pressure p for a homogeneous porous medium. 1 u = k(V'pf). Jl ( 1.3) (We have adopted f here as a term related to gravity to preserve notational consistency with [5, 6, 2]). This relationship has been put on a theoretical footing by Bear [9] via upscaling techniques applied to the conservation of momentum equation for a liquid phase. In ( 1.3), Jl denotes the viscosity of the fluid and k indicates the permeability of the porous medium. This is in general a secondorder tensor which arises via an upscaling procedure to account for our incomplete knowledge of the exact geometry of a porous medium. Darcy's law is supplemented with the conservation of mass, which may be written (under the assumptions that the medium is rigid, homogeneous, and the liquid phase is incompressible) as: V'. u = g, (1.4) g indicating source terms. Defining K := lk (called the hydraulic conductivity), J1. Darcy's problem on a domain 0 is formed using these last two equations: u + K(V'pf)= 0 inn, V'u=g on n. (1.5) The general boundary condition for this problem is au n + (3p = b. (1.6) 9
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In the case that a = 1 and {3 = 0, we are imposing Neumann boundary conditions which fix the flux, and when a = 0 and {3 = 1, we are imposing Dirichlet boundary conditions which fix the pressure. For our purposes, we will assume the former, instead fixing the constant for the pressure solution by requiring fu p = 0. 1.2.2 Simplifying assumptions Assuming that the porous medium in question is homogeneous indicates K is constant throughout the domain. Next, under the assumption that the medium is isotropic, we may write K = = !5:_I J.l J.l (I indicating the identity tensor), with K, now indicating permeability (see [9, 20]). With a := E JR.+ constant in the domain n, the Darcy problem may now be written as: Find u and p such that au+ \lp = f, \! u =ginn, ( 1.7) u. n = 0 on an. ( 1.8) Here, we have assumed homogeneous boundary conditions and that g E L2(n) satis fies the compatibility condition fn g = 0. We make the further simplification that f is a piecewise constant function. Remark 1.1 Although we will assume, for the sake of simplicity, that u n = 0, we could more generally assume u n = b on an. and require satisfaction of the compatibility condition JKg = f8ub. In [27], it is shown that 3wb E H(div,n) such that wb n = b. As such, we can recover the homogeneous case given above with the modified righthand sides f awb and g \1 wb. 10
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1.2.3 Weak form of Darcy's problem Darcy's problem shows the relationship that must exist between velocity and pressure in the case of a continuous velocity and pressure which is continuous with continuous first derivative. However, it is wellknown that the corresponding function spaces are not complete and therefore do not provide the correct setting in which to provide existence and uniqueness results. To approach this, we must turn to the variationa] form of (1.7). Formally dotting the first equation of (1.7) with v E H0(div, n) and multiplying the second equation of (1.7) by q E and integrating over n yields: r au v + r V' p v = r f v' ln ln ln ( 1.9) r V' uq = r gq. ln ln ( 1.1 0) From here, we may integrate ( 1.9) by parts and apply the condition v n = 0 on an to yield the equivalent statement r au. v r pV' v = r f v, ln ln ln fv.uq= fgq. ln ln Noting the boundary condition u n = 0, as well as the definitions ( 1.1) and (1.2) of the L2inner products, we state a variational form of ( 1.7). Find ( u, p) E Ho(div, n) X such that (au, v)n(V' v,p)n = (f, v)n 'Vv E Ho(div, f2), (V' u, q)n = (g, q)n Vq E II (1.11)
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Alternatively, this can be expressed as: Find (u,p) E H0(div, n) x such that B((u,p), (v, q)) = F(v, q) V(v, q) E H0(div, 0) x (1.12) where B((u,p), (v, q)) :=(au, v)n(\7 v,p)n(\7 u, q)n, (1.13) F(v, q) := (/, v)n(g, q)n. (1.14) We shall frequently refer to the symmetric form B( ). A fundamental question we must address when trying to develop a numerical method to approximate the solution of a problem is whether the problem has a so lution to approximate in the first place. Making use of classical theory, the results of which are listed in the appendix, we state the wellposedness of the weak Darcy problem in ( 1.12). Theorem 1.2 Problem ( 1.12) is wellposed when we assume n is starshaped (see Remark 8.5). Moreover, defining a(u, v) :=(au, v), the a priori estimates hold: 1 1 a llulldiv,n + /3(1 + ( 1.15) 1 a a a IIPIIo,n /3(1 + + (32 (1 + ( 1.16) Proof: First, note that a(u, v) = (au, v)n is coercive on Ker(\7) c Ho(div, n) by definition of the [L2(0)]2 norm. So, by Lemma B.3, there exists a> 0 such that conditions (B.2) are satisfied. Next, since we may write [HJ(O)Jd c H0(div, n), it follows from Theorem B.4 that operator \7 : H0(div, 0) t is surjective. But this implies that there exists (3 > 0 such that condition (B.3) is satisfied. So, we have proved the conditions of Theorem B. I, and so problem ( 1.12) is wellposed, and the a priori estimates follow immediately by observing llall =a. 12
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1.3 Discretization and approximation Although Theorem 1.2 assures us of a theoretical solution to the Darcy problem, it is not always clear what this solution is. We therefore tum to an approximation approach based on finite elements in order to find an approximate solution. This requires finitedimensional spaces defined on a triangulation of the domain 0. In Subsection 1.3.1, we first focus on forming these spaces and the related notation. With these points established, we define the main problem we address in this work together with a few alternatives for dealing with it in Subsection 1.3.2. 1.3.1 Notation related to discretization Let {TH} be a family of regular partitions TH of n composed of triangles K. Define Vo := {v E 2(0): viK E IP'o(K)VK E TH}, Vk := {v E C0(0) : viK E IP'k(K) 'v'K E TH, k 1}. We set V k := [Vk]2 n H0(div, 0). Next, define Qk := vk n Q_k := {q E qiK E IP'k(K)'v'K E TH}, ( 1.17) ( 1.18) ( 1.19) (1.20) the spaces of piecewisedefined polynomials which are continuous (resp. discontin uous) over 0. It will prove convenient in the following chapters to refer to these last two spaces collectively when linear interpolation is used. In this case, we refer to them as Qd. dE { 1, 1}. Denote by 8K the boundary of K E TH which consists of edges F. Denote by H the set of internal edges of TH. The area of K will be denoted by I K I We define 13
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K K' F J Figure I .2: The normal vector. HF := diamF, HK := diamK, and H := max{HK : K E TH }. It is clear that IKI :S H'f<. Also, for each F = K n K' E H (see Figure 1.2) we choose a fixed unit normal vector nF. The standard outward normal vector at the edge F with respect to the element K is denoted by np:, and coincides with nF in the case F c 80. Define HJ(TH) := {v E L2(0): viK E HJ(K)}, := {v E L2(0): viK E Ho(div, TH) := { v E L2(0) : viK E Ho(div, K)}. ( 1.21) ( 1.22) ( 1.23) Several operators acting on these spaces will be used in the discussion. Let D be some domain in JR.d, d = 1, 2, and suppose Z is a vector space defined on D. We shall denote the identity operator by I0 Given a scalar function q, its projection ng(q) into the Hilbert space Z defined on domain D will be given by (llg(q), z)o = (q, z)o 14
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V z E Z. When D = 0 and Z has a definition on all of 0, we shall take = LKETH When the projection is onto the space of constants, we shall use llD(q) := 1b1 JD q. Borrowing from [3], one denotes its jump [q] and average value {q} as respectively. We may further define the jump and average values of a vectorial quantity v by Suppose q is a function for which we can define its jump across an edge FE H. and let K, K' E TH be the two triangles incident to F. Then, for arbitrary integrable vector v, it follows, { { ( K K K' K') K [ ] K) (v,[q])F= 1Fv[q]= JF VnFq +vnFq =(vnF, q nF F (1.24) From this, we can clearly see (1.25) and so we shall interchange these expressions of the norm wherever necessary. Furthermore, the following identity holds (see, e.g., [3]) :E(q,vn)aK= :E([q],{v})F+ :E({q},[v])F, (1.26) KETH FEE:H FEE:H when we define [v] = 0 and {q} = 0 on an. Also, take [q] = 0 on an. 15
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Finally, two local operators 7rK() andNK() acting on [IP't(K)J2 are given by the expressions 7rK(vt) := L llF(Vt Fc8K NK(vt) := L llF(Vt (1.27) Fc8K where v1 E [IP't(K)j2. The function is given by, ( 1.28) being the vertex of cell K opposite to the edge F. These form the local basis of the lowest order RaviartThomas space, and we see that 7rK(v1 ) is the local Raviart Thomas interpolant of v1 Furthermore, we define the operator 1r( ) acting on V 1 such that 1r( v1 ) IK = 7rK( v1). In accordance with the relevant definitions, we conclude that 7r(vt) E Ho(div, n) is the global RaviartThomas interpolant of Vt (see [34, 4, 22]). The functions are elements of with the property V'1J: = and are given by the expression: K HFa (x x K ) 1JF = 2IKI 2xF x+CF (1.29) where CF is fixed to ensure JK = 0 (see [5, 6]). Remark 1.3 It may be shown that the operator 1r( ) is injective. 1.3.2 Numerical approximations Consider the the unstable mixed method (i.e., that which leads to a singular stiffness matrix) for the Darcy problem when discretization is carried out with the IP\/IP'1 element for both continuous and discontinuous pressures. Find (u1,pt) E V1 x Qd such that ( 1.30) 16
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2 10 105 1010 1015 10010 102S 10"' ; 0 10.16 10_.. 10"' 1020 1010 10 0 200 oo 600 800 1000 1200 Figure 1.3: The lefthand plot shows how the singular values are distributed, with the xaxis showing the values (no meaning is assigned to theyaxis). The righthand plot shows the value of the singular values along the yaxis. In this case, the xaxis indicates the position of the singular values in a vector ordering them from largest to smallest, with a total of 1023. for all (v1 q1 ) E V1 x Qd. Figure 1.3 displays plots of the singular values of the stiffness matrix formed from method 1.30 using discontinuous pressure interpolation for a particular instance of the "crisscross" mesh, similar to that shown in Figure 1.4. Of the I ,023 singular values associated with the stiffness matrix, we observe a large portion are numerically zero, indicating a large nullspace. In order to implement the the associated elements for the Darcy problem, it is essential to handle this nontrivial null space. Assuming an efficient description of a basis for the nullspace, one possible approach to finding a solution to the method could be to solve the resulting system in 17
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the orthogonal complement to the nullspace. A different approach to handling the nullspace is to stabilize the method through the addition of extra terms. Approaches which follow this philosophy are called stabilized finite element methods. The methods in this work can be seen as an instance of the class of stabilized finite element methods, and so we present some related methods here for completeness. In [29], this formulation (among all other possible orders of discretization for velocity and pressure) was stabilized with the terms Here, (3 is a positive constant. Also, H.1. and (a} are defined as, H IKI + IK'I d .1. .H an 2 F (a}:= a+ a', 2 (1.32) where a is the value of a as seen from K and a' is the value as seen from K' (for reference, see Figure 1.2). Furthermore, in the case of a continuous pressure interpolation, the method in [II] stabilizes the formulation (again, among other choices for interpolation) with the leastsquares term (1.33) where projection II0 indicates the projection into the space of piecewise discontinuous constants on the domain n, a is a positive constant, and L is a characteristic length. Finally, we note one of the PGEM methods designed to stabilize the 1P't/IP'0 ele ment for the Darcy problem (1.7) (see [6]) is: Find (u1 p0 ) E V1 x Qo such that (1.34) 18
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This method was derived by enriching the velocity and pressure spaces, and in dicates an enriched pressure solution, Here, 'TJF is defined in terms of TJ!f ( 1.29) by: { TJ!f nF, TJF(x) = 0, ( 1.35) X E K and F c oK X E K and F ct. a K. Numerical experiments have validated the expected convergence properties of the method. Moreover, we have seen that for the family of meshes shown in Figure 1.4, pressure PH is actually "superconvergent" (in the sense that we see a higher rate of convergence than expected with the piecewise constant interpolated pressure without increasing the number of degrees of freedom in the problem). Due to the nice properties of the PGEM for the IP\ /IP0 element, the following work focuses on extending the underlying philosophy to the JP>IfJP>1 element for both continuous and discontinuous pressure interpolations. This will be done borrowing from the notion of consistently stabilizing as in [29], using terms which involve projections into piecewise discontinuous spaces as in [ 11 ], all within an enrichment framework driven by residuals as in, e.g. [6]. Remark 1.4 A small note regarding the tenn "stability" should be presented. The use of the word "stable" is overloaded in the framework of finite element methods. In some cases, the term refers to the sensitivity of the method with respect to the data controlling the relative size of certain terms present in the method. In limiting cases for these data, the methods may yield approximate solutions which exhibit non19
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10 0.9 0.8 101 I 0.7 )' 0.8 0.5 10. 0.4 0.3 10.1V' 02 0.1 0.2 0.4 0.6 0.8 10. 10"' 10' Figure 1.4: A sample crisscross mesh and a curve showing better than 0( H2 ) con vergence for PH in the 2norm on a family of this type of mesh. physical oscillations arising from the fact that the system matrix is nearly numerically singular. In other cases, such as with mixed methods, the term "stable" may refer to the invertibility of the matrix when formed using different combinations of approximation spaces to discretize the fields which are to be approximated. In this case, "instability" is not numerical in nature, but reflects a fundamental incompatibility of the spaces as described by the famous LBB condition (see, e.g., [ 16 ]). This latter form of instability is the one we address in this work. 20
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2. The RELP methods In this chapter, we introduce several methods by which we obtain approximate solutions of the Darcy problem. The methods all rely upon the same strategy for their development, and as such are derived in a common framework. In Chapter 3, we shall see that this new approach to deriving the methods will provide a good basis for performing a numerical analysis of the methods. In a general sense, this framework begins with an enrichment of the linear spaces for approximating both the velocity and pressure, and we derive the methods with the philosophy that the enrichment functions solve local problems driven by local projections of residuals. For this reason, we call these methods Residual Local Projection, or RELP, methods. The outline of this chapter is as follows: Preliminary definitions and results are introduced in Section 2.1. Using an enrichment of the polynomial spaces composed of linear velocities and pressures, two "general" methods are derived in the course of sections 2.22.4. In particular, we will motivate the need for, and requirements on, the enriching spaces in Section 2.2. In Section 2.3, we will use these motivations to explicitly define these spaces. In Section 2.4, these spaces are used to derive the methods, which are presented in Subsection 2.4.3. The two general methods are shown to have unique solutions in Section 2.5. In Section 2.6, we derive four methods based upon the general methods of Subsection 2.4.3. 21
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2.1 Preliminaries We will make use of several nonstandard operators and spaces in the sequel, and we focus on their definitions in this section. First, a set of operators acting on the space V 1 will be defined in Subsection 2.1.1. We will then define a few subs paces of H0(div, 0.) and in Subsection 2.1.2. Lemma 2.2 of this subsection will be important to the derivation presented in Section 2.4. 2.1.1 A few operators A set of operators will be required throughout this work. We begin with their definitions and establish a few properties relating the spaces we will be considering. First, denote by vF( v nF) the operator which stands for either IF( v nF) := v nF or llF{ v nF ). We make use ofthis to define operators PK( )and PK( )acting on functions v1 E [IP\ (K)J2 and having values in H(div, K) and respectively: (2.1) We see first of all that the functions PK( v1 ) defined in this way satisfy the compatibil ity condition JK '\1 PK(vi) = faK PK(vt) nK, and so problem (2.1) is wellposed. Global versions p( ) and P ( ) acting on v 1 E V 1 are deft ned such that p( v 1) I K = PK(vl) and 'P{v1)IK = PK(vt). It is clear that p(v1 ) E H0(div, 0.) as p{v1 ) shares with v1 the continuity of their normal components. We make a few important observations regarding problem (2.1 ). First, if (2.2) 22
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it is easy to verify [5, 2, 6] that (2.3) (where 7rK and NK are defined in ( 1.27)), and we conclude that (2.1) has an analytical solution in this case. On the other hand, a numerical method is required to approximate the solution to (2.1 ), if (2.4) For the time being, we leave the strict definition of vF alone. However, the choice of either (2.2) or (2.4) will ultimately lead to one of two branches of numerical methods. This will be explored in more detail in the sequel. Remark 2.1 It may be shown that when PK() : IP\(K) + PK(IP\(K)) satisfies the boundary condition PK( vl) n = v1 n, then it is injective, and we conclude that the global operator p( ) is injective as well. See Lemma 4.3. 2.1.2 A few subspaces As has been indicated, we will derive methods based on residuals. We will fix notation related to these residuals below. Given arbitrary functions v1 E V1 and q1 E Qd and fixed functions f and g, we may define the residuals related to the Darcy problem, We would like to define a subspace of base upon RM (v1 q1 f). Its definition will require the following problem of finding q E '1:/K E TH such that, D.q = 'V RM (v1 q1 f) inK, (2.6) 'VqnF = vF(RM(vi,qi,f))nF on F C 8K. 23
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Noting that J K /j.q = faK \7 q n, problem (2.6) is wellposed. Letting wK(v1 q1 f) denote the formal inversion operator for this problem, we also consider a global form w( v1 q1 f) of this operator with the property w(v1, q1 f)IK = wK(v 1 q11 f). Assuming f is fixed, we define the following finitedimensional space: Rather, we see W is composed of all solutions to problem (2.6). We denote the orthogonal complement of Win by W_l_. Let p(V 1 ) to be the image of the space V 1 under the operator p( ). The next Lemma presents important information regarding the relationship between some of the function spaces we have defined. Lemma2.2 QdnW_l_ = {0} and p(VI)nH0(div,TH) = {0}. (2.7) Proof: First, suppose q E Qd n Wj_. Then (q, t) = OVt E W. Next, select i E W such that ilK = wK(O, q, 0), rather, i satisfies the local problems for each K E TH. fj.i = V' RM (0, q, 0) = 0 inK, (2.8) A M V't nF = IIF(R (0, q, 0)) nF = \i'q nF on F C 8K. (2.9) The unique solution i to this problem, since q E W j_ c ), has the property ilK = qiK IIK(q) = qiK It follows then that 0 = (q, i) = EKETH and so q = 0. 24
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Next, suppose v E p(Vt) n H0(div, TH) Since v E p(Vt), it follows that there exists v1 E V1 such that v = p(v1 ) and P(v1 ) satisfy the local problems (2.10) Since v E Ho(div, TH ), v nF = p(v1 ) nF = 0 for each FE t:h. Moreover, since 'V viK = 'V PK(vt) = 'V vtiK is constant, the divergence theorem applied on each K E TH tells us: \1 VIK = 1 r \1 V IKI JK 1 r K = IKI laK v. n =0. So, in fact (2.10) has zero righthand sides, indicating v = p( v1 ) = 0. 2.2 The enriching approach: Motivation As the linear spaces defined in the previous section have the properties: Vt C Ho(div, 0), (2.11) a natural starting point for approximating the solutions ( u, p) to problem ( 1.12) is: Find (u1,p1 ) E V1 x Qd such that (2.12) for all v1 E V 1 and q1 E Qd. Unfortunately, it is wellknown that f (V' Vt, qt)n 0 m sup = q1EQd v1EV1 llqtllo,nllvtlldiv,n (2.13) and it is clear that condition (8.3) of Theorem B.l is not satisfied. In this case, the issue is that the space V1 is not "large" enough for V': V1 + Qd to be surjective. 25
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Remark 2.3 See [22] for an explanation of the famous checkerboard mode, which shows the result in Equation (2.13). Our goal is to derive methods which overcome this incompatibility of spaces by sup plementing the linear spaces of problem (2.12) with spaces of functions that contain the information necessary to satisfy the LBB condition. We will derive the methods based upon the PGEM approach [2, 6, 5]. To begin, we assume trial spaces which are as large as possible. Thus, we choose the trial spaces V :=VI+ Ho(div, 0) and Q := Qd + (Note, we could simply write H0(div, 0) and (0), but we choose the above nota tion to emphasize that we want our trial functions to retain information about degrees of freedom related to VI and Qd.) Ultimately, we will choose suitable subspaces of these trial spaces. For now, consider the following multiscale decomposition of the functions in these spaces: write (vH, QH) = (vi + Ve, QI + Qe), where VI E Ve E H0(div, 0), QI E Qd. and Qe E (0). As for the test space, we will take it to be as large and as local as possible such that ( v H, QH) decompose uniquely as (vi + vb, QI + Qb). With this in mind, we choose for the velocity and pressure test spaces, respectively: p(V I) EB H0 ( div, TH), (2.14) This choice is motivated by Lemma 2.2, a result which will allow us to use the static condensation procedure. We shall consistently refer to Ue, Pe vb, and Pb as enrich26
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ment functions. We now modify problem (2.12) using these updated spaces and define the fol lowing PetrovGalerkin method: Find ( UH, PH) E V x Q such that (2.15) for all VH = p(vi) + Vb E p(VI) EEl Ho(div, TH) and QH = Ql + Qb E Qd EEl w..L. We must begin by asking the important question as to whether the problem (2.15) is wellposed. In fact, the answer up to this point is no. However, we investigate the reason for this in the sequel and rectify the problems using the new enriching strategy. We refer to Theorems 8.6 and 8.7 in the appendix to provide the framework needed to investigate this issue. We may write (2.15) as: Find ( uH, PH) E V x Q such that (2.16) Defining C V': p(V1 ) EEl H0(div, TH) Qd + B := V': V1 + H0(div, TH) Qd E9 W..L, and A := I : Ker(B) Ker(C), we see this fits the framework considered in Theorem 8.6. There are two things we need to note. 6T is not injective since Qd + is too large. Consider the related term where we have used the constant value of V' v1 to establish (V' v1 Pe)n = 0. First, since V': H0(div, K)is surjective, it follows that the last term on the righthand side above controls Peon each K, and this indicates that Pe is controlled on all of n. We now question what control over p1 exists. As 27
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previously noted, V 1 and Qd are incompatible spaces, so we do not expect to gain control over p1 with the term (V' v1,p1)n. In fact, we may see where the deficiency arises by considering that there exist positive constants C1 and C2 such that (See [26] for details), So, in order to control Ph we must control V'p1 and Therefore, theresponsibility of controlling these values falls to vb through the term ('\7 vb, p1 )u. However, integration by parts and the fact that vb nF = 0 for each F E t:H indicates L ('Vvb,Pl)K =L [(vb, 'Vpl)K + L (vbn:,pl)Fl KETH KETH FC8K =L (vb, 'Vpl)K (2.18) KETH This makes clear that the term ('\7 vb, p1 )n accounts for the deficiency in infor mation regarding V'p1 because for each K E TH there is a vbiK E H 0 (div, K) such that 'VvbiK = =IP1 (p1IKIIK(pl)), meaning that substitution and the u K inverse inequality imply However, the term cannot account for the deficiency in information regarding A is not an isomorphism since Ho(div, n) (and importantly, Ker(B)) is too large. 28
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Remark 2.4 Problem (2.16) indicates in particular that (\7 uH, Qb)n = (g, Qb)nfor all Qb E Wl.. Rather, (\7 Ue, Qb)K = (Rc, Qb)K for all Qb E Wl. and K E Tn, leading us to conclude (\7 ue) = (Rc). We see that in order to have a wellposed problem, we need to restrict the func tions Ue and Pe to subspaces of H0(div, 0) and respectively. We motivate the choice of these subspaces with the following points: I. Define subs paces of the trial spaces we have chosen in problem (2.15) for which the formulation is conforming and wellposed; 2. Yield enrichment solutions which are residualbased. 3. Don't increase the number of degrees of freedom over those present in the residuals. 2.3 The enriching approach: Defining appropriate subspaces In this section, points 13 at the end of the last section will be addressed in terms of the discussion related to operators 6 and >A. First, Theorem B.7 suggests that enforcing Ue E H0(div, Tn) (rather, enforcing zeroflux boundary conditions for functions Ue on the edges of the triangulation), just as is done for the enriching por tions of the velocity test functions, will ensure that >A is an isomorphism. However, we do not want to enforce Ue nF = 0 on the edges F since doing so will destroy the surjectivity of operator iJ through a loss of control over the jumps of q1 Instead, with an eye on keeping the method conforming and residual based, we enforce for each FE H 29
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where o: > 0 is some real number independent of H F and a. This choice is equivalent to imposing homogeneous boundary conditions since nonzero boundary conditions are accounted for with a perturbed righthand side when solving for a function in Ho(div, TH) (See the remark after problem (1.7)(1.8)). Enforcing ;;F IIF([p1D) nF will replace into the method the information lost according to the comment on the nontrivial kernel of 6T. We hope this choice renders problem (2.15) wellposed, up to \1 Ue E Wl.. Note that Remark 2.4 indicates that rrw.L('Vue) = rrw.L(Rc), but doesn't resolve the value beyond this. So, we are permitted (and required) to choose the value of rrw+JR('V ue) This motivates our choice for the subspace of H 0 (div, 0) we seek. Again, with an eye on stabilizing the method using residuals, and taking care to ensure the method is conforming and that the compatibility condition JK 'Vue = faK UenK is satisfied, we choose Ue to come from the space: (2.21) where Here, CK is a constant meant to ensure compatibility with the boundary condition ;;FrrF([qtD) nF and has definition, (2.22) 30
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Having addressed the points regarding C and A from Section 2.2 within Ve, we now take for the enrichment pressure space (2.23) With these definitions we have determined the spaces in which we seek a solution to problem (2.15). We shall derive the numerical methods to be considered here and show that they are wellposed. 2.4 The enriching approach: Derivation of the methods The problem we now wish to consider is: Find (uH,PH) E (VI+ Ve) x (Qd+Qe) such that B((uH,PH ), (vu, QH )) = F(vH, QH ), (2.24) for all VH = p(vi) + Vb E p(Vt) EB vb and QH = Ql + Qb E Qd EB W_l_. Consider the problem in an equivalent form, which we obtain through a standard static condensation procedure: B((ut + Ue,pi), (p(vi), Qt) = F(p(vt), Qt) \f (p(vt), Qt) E p(Vt) x Qd, (2.25) (auH, Vb)K(pH, \7 Vb)K(qb, \7 UH )K = (/, Vb)K(g, Qb)K (2.26) \f(vb,Qb) E Ho(div,TH) X w_l_. Problem (2.24) is wellposed if and only if the system (2.25), (2.26) is well posed. In this system, (2.25) represents the method and (2.26) represents local prob lems. This section will focus on three points related to system (2.25) and (2.26). First, we will discuss the structure of the local problems and develop a notion of how their solutions will affect the method. We then describe solutions to the local problems. 31
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Finally, we will finnly define two general methods by substituting the local solutions into the (2.25). 2.4.1 Local problems in the method The local problems may be written as: (2.27) From now on, when it is clear that RM(w1,r1,f) and Rc(w1,g) depend on the solutions ( u1 p1 ) to method (2.25), we shall shorten the notation of these to RM and Rc, respectively. Using the definition of Ve. this local problem, under the appropriate assumptions, is equivalent to the following strong local problem: inK, (2.28) on F C 8K, for each K E TH. Motivated by the linearity of this problem, we define the functions and by the problems, VvM = 0 e inK, (2.29) on F C 8K, a + V = 0 in K G C C 1 Vve = R (v1,g)llK(R (v1,g))H2 w(vi,qi,/) inK, a K on F C 8K, (2.30) 32
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and, on F c 8K. (2.31) Then we see the solutions ( ue, Pe) of problem (2.28) satisfy Ue = + uf + uf and Pe = + pf + pf, where (uf,pf), (uf,pf) satisfy problems (2.29), (2.30), and (2.31) with V1 3 v1 = u1 and Qd 3 q1 = respectively. Remark 2.5 It is important to remark that the conditions on Ue nF satisfy the compatibility condition given by the divergence theorem. First, we see that either of the possible two values for vF(RM) satisfy the compatibility condition for Indeed, for either choice: r L r(RMvF(RM))nF laK FC8KJF = L 1 (RMvF(RM)) nF Fc8K F = L ( r RMr vF(RM)) nF Fc8K JF JF =0 = i Also, uf satisfies the compatibility condition: (2.32) 33
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Finally, since w( u1 p1 /) IK E ( K), it is clear the compatibility condition is satisfied in problem (2.30). Remark 2.6 It is important to note that (ue,Pe) (and also (uf,pf), and are defined globally in H0(div,n) x through the local problem (2.28), which they satisfy for each K E TH. This follows because problem (2.20) ensures [ue] = [uf] = = = 0. It is now clear that problems (2.29)(2.31) are wellposed and define an enrich ment solution ( Ue, Pe) to (2.27) in terms of the linear portion of the solution ( u1 pi). Having considered the significance of local problem (2.26) we now consider method (2.25), which may be rewritten as L + op(vt))K(\7 + qt)K] KETH We may rewrite (2.33) in an equivalent form. First, problem (2.1), integration by parts, and using nF = 0 for each FE [H gives us, ap(vt))K = \7PK(vt))K = (\7 PK(vt))K, and we conclude, using \7 E for all K E TH, L KETH = L (\7 PK(vt)ql + IIK(qt))K (2.34) KETH 34
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Furthennore, integration by parts also shows us, and we conclude from this, the fact = 0 on all internal edges, and the identity (1.26), (2.35) FEl:H Substituting tenns (2.34) and (2.35) into equation (2.33), we see the method is described by the equation, B((ut + (p(vt),qt) + L (\7 PK(vt)q1 + ITK(qt))K (2.36) KETH Written in this fonn, it is clear an implementation of the method requires explicit definitions for the functions and However, we need only include \7 from problem (2.30), for which we currently have all tenns, aside from w(ut,Pt, f). We now tum to either giving analytic expressions for, or at least characterizing, these enrichment solutions. 2.4.2 Solutions to the local problems First, consider problem (2.31 ), which satisfies for each K E TH, = 0, \7 = l:l a L IIF([pt]) FcoK inK, on F c 8K. 35
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Denote the fonnal inversion operator of this problem by and let := and 'Di(([p 1 ]) := Following the approach taken in [5, 2, 6], the solution to this problem is given by: Dl """"' aiJF([pt]) K ue K L....., H r.p F and Fc8K FO' (2.37) Dl """"' aiJF([pt]) K Pe K L....., H 'TIF' Fc8K FO' (2.38) where and are given in ( 1.28) and ( 1.29), respectively. Next, consider problem (2.29) auM + 'VpM = RM 'VuM = 0 e e e inK, on F C 8K. Denote the fonnal inversion operator of this problem by Due to the assumption that f is piecewise constant, as is 'Vpr. we conclude that (2.39) We now address the portion of the solution := MK( aut) It is clear that inK, (2.40) on8K. Comparing to problem (2.1), we see that = PK(u1 ) and PK(ut)Ut. 36
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Remark 2.7 lnfact, define := and aMi<(u1 ) := Then, since V ( = a (u1 aMi<(u1)) = a(IK aMi( )(ut), we see that PK(ut) = = and PK(ut) = (IKaMi< )(ut) = Ut Using this notation, we see the solution to problem (2.29) is Remark 2.8 In fact, this solution is only explicit up to the definition of the operator PK(ut). which is itself dependent on the explicit choice of boundary condition vF(). We discuss the implications of the choice momentarily. We are not able to derive analytical solutions for as defined in problem (2.30). However, this is not an issue from the point of view of the implementation of the method (2.36), as only V must be included. We know the value of this from problem (2.30). If required for postprocessing, this may be approximated by twolevel methods following [2]. We also note that (2.42) This may be seen by noting that problem (2.29) implies satisfies: VRM e (2.43) and comparing to problem (2.6). Taking the solutions to equations (2.29)(2.31) together, we conclude that UeiK = (PKIK )(ut) + + Vj(([pt]), (2.44) PeiK = PK(ut) + (IKllK )(/XPt)IK + + (2.45) 37
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2.4.3 Two general methods Substituting the solutions for and uf into (2.36) and noting the injectivity of the global operator p( ) (see Remarks 1.3 and 2.1 ), the method takes on the form: Find (ut,pt) E V1 x Qd such that (2.46) B9((ut,Pt), (vt,Qt)) := B((p(ui),pt), (p(vt),qt)) + L apK(vt) + 'Vqt)K L (2.47) FECH F F9(vi, QI) := (!, p(vt))uL (rrK(g) + (IKITK)(f x), Q1) KETH K K L (gf 0 x, PK(Vt)) 0 (2.48) KETH K K We have now arrived at a final form for what shall hereafter be referred to as the "general method." We also introduce a reduced form of this method. This reduced form is obtained by neglecting the terms 'KETH(uf,ap(v1 ) + 'Vq1)K. We shall see that doing so doesn't undermine the stability or convergence rates of the general finite element 38
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method and has the advantage of yielding a method which is symmetric. Taking, ( Ut, Pt), ( Vt, Qt)) := B( (p( Ut), pt), (p( Vt), Qt)) 1 L aH2 (PK(ut)Pt + ITK(pt), PK(vt)Qt + ITK(qt))K KETH K L (IIF([pt]), IIF([qt]))F, (2.49) FECH a F the method is: Find ( u1 p1 ) E V 1 x Q d such that, (2.50) for all ( v1 qt) E V 1 x Qd. This method shall hereafter be referred to as the "symmetric general method." Remark 2.9 Note that we have dropped the term because \1 u1 ITK (\1 u1 ) = 0. However, we continue to regard this term as present in the methods. This is particularly important when the consistency of the methods is proved in the next chapter. Remark 2.10 Although we shall perform the error analysis for the methods as given above, the term, L (g, PK(vt))K, (2.51) KETH may be dropped when performing computations. In Section 3.4.3, this term is shown to be at order H2 and as such neglecting it does not undermine the convergence properties of the method. 39
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2.5 Existence and Uniqueness Results We must now show the methods (2.46) and (2.50) presented in Section 2.4 have unique solutions. We begin by defining the meshdependent norm: [ 2 """' 1 2 """' 1 2 ] 1/2 lll(u,p)IIIH := allullo,n + ;:IIY'PIIo,K + a H)IIIF([p])llo,F (2.52) KETH FEt:H The methods will be shown to satisfy an infsup condition in this norm in Subsections 2.5.2 and 2.5.3. The theory of these two subsections requires a few preliminary results, which can be found in Subsection 2.5.1. 2.5.1 Some auxiliary results We now establish some auxiliary estimates on terms appearing in the method due to enrichment. The overall goal of these results is to bound terms present in methods (2.46) and (2.50). This will assist us in establishing the results of Subsections 2.5.2 and 2.5.3, as well as the convergence theory in the following chapters. Lemma 2.11 Let K E TH. Then there is a constant C1 from the inverse inequality such that (2.53) and (2.54) Proof: The first of these results follows directly from the inverse inequality of Lemma A.3 on the one hand, and the Poincare inequality A.2 on the other, noting thatp1 IIK(Pd E The second follows by observing PK(ul) E and applying the Poincare inequality. 40
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In particular, result (2.53) of Lemma 2.11 establishes an equivalence between a terms present in the methods and one of the key terms we seek to control according to equation 2.17. Now, note the following lemma Lemma 2.12 For each K E TH, let z E L2(8K) and constant DK defined on K satisfy the compatibility condition JK DK = faK z. Further, suppose (v, q) E H(div, K) X n H2(K)] satisfies the local problem av + 'V q = 0, 'V v = D K in K, v nF = z on 8K. Then there exist constants C1 and C2 independent of H K and a such that (2.55) Proof: From the definition of the 2 norm, integration by parts, the fact q E CauchySchwarz' inequality, and the local trace inequality (A.2), and the Poincare inequality, we have = ('Vq, 'Vq)K = (q, /:::,.q)K + (q, '\Jq nK)aK = a(q, 'V v)K + (q, 'Vq nK)aK = (q, 'Vq nK)aK :::; llqllo,aKII'Vq nKllo,aK :::; Ct + HKlqli_K) 112II'Vq nKllo,aK :::; J2CtH:f2II'Vqllo,KII'Vq nKIIo,aK = J2CtH:Pll'Vqllo,Kallzllo,K 41
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from which we conclude the second inequality of (2.55). To see the first inequality, consider that the trace inequality (A.3) tells us = IIVq :S Ct + HKIVqli,K) < Ct(CI + 1) IIV 112 HK q O,K This establishes the first inequality of (2.55). The previous lemma is useful in establishing a few bounds for as in the following corollary, which bounds its norm in terms of norms on its local boundary conditions. Corollary 2.13 Suppose is the pressure solution given by problem (2.31 ). Then there exist constants c3 and c4 independent of HK and a such that Proof: We first note that is a polynomial, so each of the results in Lemma 2.12 may be used. The Lemma implies, using mesh regularity, L II llF([pi])II2 :S :S L II nF([pi])II2 FC8K F O,F Fc8K F O,F L :S :S L Fc8K F FcaK F The theorem follows from the definitions of cl and c2 from Lemma 2.12. Remark 2.14 We may define a global version of Corollary 2.13 by observing (2.56) 42
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2.5.2 Existence of a unique solution for the symmetric general method Lemma 2.15 Let .) be the bilinear form defined in (2.49). Then there exists a positive constant /3. independent of H and a, such that and we conclude that, sup qt), (wt, rt)) :2: /3lll(p(vi), qi)IIIH, (wl,rl)EV1 xQr{O} lll(p( Wt), rt)IIIH forall(vt,qt) E Yt x Qd. Proof: Using the definition (2.49) of and Lemma 2.11, where we set,B = min{J1 H The last result foltows by noting that m(p(vt), qt)IIIH = 43
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Remark 2.16 Since the choice a = 1 is sufficient for the solution to exist, we will make this assumption on the value of a whenever a symmetric method is considered. 2.5.3 Existence of a unique solution for the general method Lemma 2.17 Let B9(., .) be the bilinear form defined in (2.46). There exists C > 0 such that if a C, then there exists a positive constant {3, independent of H and a, such that and we conclude that, Proof: First, observe that we may write L \lqi)K. (2.58) KETH Equation (2.57) already establishes a bound for (v1 q1)), so we tum to the other two terms on the righthand side. First, from the local problem (2.31), Young's inequality, and Corollary 2.13, 44
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where C4 = 4CrCr. Again, using problem (2.31 ), Young's inequality and Corollary 2.13, we have, (2.59) Using these two bounds and Equation (2.57), we find there are positive constants /I and 12 such that Taking /l = 4/3,12 = CJ, and assuming a :S: 204
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2.6 Explicit methods formed from the general methods All that remains is to form explicit methods from the general methods by fixing the boundary condition in the problem we solve for = (PKIK) ( u1 ) in problem (2.29). This in tum requires us to fix the boundary condition for PK(u1 ) in problem (2.1 ). As indicated previously, two choices are possible, and so a total of four methods are presented. In the case of either choice, we note that whatever analysis we are able to perform on the general and symmetric general methods that doesn't require a specific boundary condition is inherited. In particular, the methods considered in this section are wellposed owing to Lemmas 2.15 and 2.17. One choice, vF(au1 ) := IIF(aui), leads to a closed form for the function p(ui), and therefore a closed form for Methods arising from this choice are presented in Subsection 2.6.1. For the other, vF(au1 ) := the analytic solution is not clear. Thus, we need to approximate p( u1 ), leading us to develop a twolevel approach to solving the method. These methods are presented in Subsection 2.6.2. 2.6.1 Methods with analyticallydefined terms Assume that we want to solve problem (2.1) with vF(au1nF) = IIF(au1nF) This yields the problem (2.62) (2.63) We recall that (2.3) tells us PK(ui) = 7rK(u 1 ) and PK(u1 ) = NK(ui) (i.e., the local RaviartThomas interpolant of u1 and its potential). From this, we conclude 46
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We see that the general method (2.46) takes the form: Find (ui,pt) E VI x Qd such that B1((ui,PI), (vi,qt)) := B((1r(ut),pt), (1r(vi),qt)) + L (V}(([pi]),a7rK(vi) + \7qi)K (2.65) L (nF([pi]), nF([qi]))F, (2.66) FECH a F FI(vi, qi) := (/, 7r(vd)nL (nK(g) + (IKnK )(/. x), qi) KETH K K (2.67) Following the symmetric general method presented in (2.50), we define a sym metric version of method (2.65): Find ( ui, PI) E VI x Qd such that B!((ui,pt), (vi, qi)) := B((7r(ui),pt), (7r(vi), qi)) 1 L aH2 (NK(ui)PI+ llK(PI),NK(vi)qi + nK(qi))K KETH K (2.68) I: (nF([pi]), nF([qi]))F (2.69) FECH a F As we know the exact expressions for 7rK(ui) and NK(ui). methods (2.65) and (2.68) are fully discrete. 47
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2.6.2 Methods requiring a twolevel solution In this section, we assume the operator PK ( ut) is defined by problem (2.1) with vF(au 1 ) = au1 This yields the problem PK(ut) + \l'PK(ut) = 0, \1 PK(ul) = \1 u1 inK (2.70) PK(ul) nF = u1 nF on 8K. Notice the term (/, p(vd)K present in the righthand side of methods (2.46) and (2.50). Recalling that PK(vi) = v1 + f = \lM'l<(f), and problem (2.29) (which defines we note the following: (/, PK(vl))K = (/, v1 + = (/, vt)K + (\1 M'l<(f), = (/, V1)K (M'l<(f), \1 + (M'l<(f), nK)aK = (/, vl)K (2.71) So, for this particular operator p( ), we may replace the term (f, p( v 1)) on the right hand side of (2.46) with the term (f, v1). We next observe that problem (2.70) does not have a closedform solution, and so we must solve via a numerical method. Taking (pK(ui), PK(u1)) to be an approx imate piecewise polynomial solution to problem (2.70), we see the general method (2.46) (with the alternative righthand side) takes the form: Find (u1,pi) E V1 x Qd such that (2.72) 48
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B 2(( Ut, Pt), ( Vt, Qt)) := B( (p( Ut), Pt), (p( Vt), Qt)) + L (VK([pt]), apK(vt) + 'Vqt)K L (PK(ut)Pt + nK(pt), PK(vt)Qt + nK(qt)) K KETH K L (nF([pt]), nF([qt]))F, FECH (J F F2 (vt, qt) := (!, Vt)nL (nK(g) + (IKnK )(!. x), Ql) KETH K K L (gJ. x, PK(Vt)) (2.73) KETH K K Again, following the symmetric general method presented in (2.50), we define a symmetric version of method (2.72): Find (u1,p1 ) E V1 x Qd such that (2.74) (vt, qt)) := B((,D(ut),pt), (p(vt), Qt)) L (PK(ut)Pt + nK(Pt), PK(vt)Q1 + nK(qt)) K KETH K L (nF([pt]), nF([qt]))F, FECH (J F In Chapter 4, we make the terms PK(u1 ) and PK(u1 ) explicit and provide an analysis for this fully discrete method using an approximation approach presented in [2]. It should be noted that as we solve for PK(ut) and PK(ut) locally, methods (2.72) and (2.74) are twolevel methods. 49
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3. Error Analysis of the General Methods We focus in this chapter on providing an analysis of the general methods (2.46) and (2.50). Since these results apply for either choice of boundary condition vF( u1 n) (See Subsection 2.1.1 ), this analysis will apply to any of the related methods in the case that the expression for PK(u1 ) is known exactly. Therefore, methods (2.65) and (2.68), for which PK( u1 ) = 7rK( u1 ), automatically inherit the analysis of this chap ter. However, methods (2.72) and (2.74) involve approximations to PK(u1). This approximation will introduce extra errors into their formulations which are not con sidered in this analysis, and so this analysis does not apply directly to these methods. These errors will be handled in Chapter 4. The outline of this chapter is as follows: In Section 3.1, we build upon standard interpolation results in order to bound the interpolation error in the mesh dependent norm (2.52). Other important bounds will be developed or cited for their use in later parts of the analysis. Consistency results for both general methods are discussed in Section 3.2. In Section 3.4, we present error estimates for both general methods presented Section 2.4. Estimates are given in both the meshdependent norm (2.52) and in natural norms. In the case a discontinuous pressure is used, the mass conser vation properties of Section 3.3 are used in the proofs of these results for the velocity. 50
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In the following, we assume C, C1o C2 denote generic positive constants, independent of H or a, with values that may vary in each occurrence. 3.1 Interpolation results Suppose u E [Ht(n)j2, where 0 ::; t ::; 2. We will denote by v1 the Clement interpolant of u. The Clement interpolation operator (cf. [27, 22]) CH : H1(n) vl (with the obvious extension to vectorvalued functions), satisfies, for all K E TH and and a stability condition, \fq E Ht(wK), \fq E Ht(wF), (3.1) (3.2) (3.3) fort= 1, 2, m = 0, 1, where WK = {K' E TH: K n K' # 0} and WF = {K E TH: KnF#0}. Next, suppose q E Ht(n) n (0). In order to take into account the approxima tion of the pressure, we consider the 2(0) projection onto Qd which is denoted by : 2(0) Qd. This projection satisfies (cf. [22]) (3.4) for 0 ::; m ::; t ::; 2. We will typically denote the interpolant by q1 We wish to use these to find a bound on the interpolation errors in the meshdependent norm. For this, we will need a bound on the norm of the jump of the difference between p and its interpolant q1 51
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Lemma 3.1 Letting q1 E Qd be an interpolant of p E Ht(O), 1 :::; t :::; 2, which satisfies the approximation property (3.4), then there exists C > 0 such that Proof: From the shaperegularity of the mesh, the definition of the jump, the trace inequality, and (3.4), we can make the following bounds and the result of the theorem is established taking the square root of both sides. We will ultimately need a bound on L:FEH which is present in the meshdependent norm. Consider the following lemma, which is the first step in establishing this. Lemma 3.2 Given q E L5(0) n H2(TH) there exist constants C5 and C6 such that Proof: The first inequality follows directly from the stability of the projection onto constants (see the appendix) using each edge F as a domain, taking C5 = ( 2 To 52
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establish the righthand inequality, consider the following local auxiliary problems: au+ '\lp = 0, '\1. it = CK in K, it nF = nF on OK. HFa au+ '\lp = 0, 'V' u = CK inK, where CK = L:FcaK is given in Equation (2.22). It may be established that L ('\1 it, q)K =L (it, 'V'q)K + L H:a KETH KETH FECH Likewise, we see L ('V' u,q)K =L (u, 'V'q)K + L KETH KETH FECH From this and the fact 'V' it = 'V' u, it is clear 53
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We need a bound on L:KETH (itu, '\1 q) K in order to establish the result. To this end, observe the Cauchy inequality, Lemma 2.12, mesh regularity, and Equation (3.4 ), So, using Equation (2.56) (with rather than we get L (itU, '\lq)K::; L + L KETH FEt:H KETH Taking C6 = CiCrCp. we get the inequality we seek. Lemma 3.3 Under the assumptions of Lemmas 3.1 and 3.2, and given q E Ht(K), it follows that (3.6) for 1 ::; t ::; 2. Finally, we may establish the result on the interpolation error as measured in the meshdependent norm. Using Corollary 3.3, we can establish an interpolation result in the meshdependent norm. 54
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Lemma 3.4 Suppose that (u,p) E ([H1(0)]2nH0(div, 0)) x is the solution to ( 1.12) and that ( v1 q1 ) E V 1 x Q1 are interpolants satisfying Equations (3.1) and (3.4 ). Then, there exists C > 0 such that I lli(uqi)IIIH :::; CH ( aiuli,u + 1 : a 2 (3.7) Proof: The result follows directly from the assumptions in Equations (3.1) and (3.4) and from the result of Lemma 3.3. In addition to these interpolation results, it will be useful to establish some results regarding the operator PK(). Since there are two such operators, we will note results for each independently and then collect the properties which are shared. Recalling that 1r( v) is used to denote the global RaviartThomas interpolation, consider the classical interpolation estimate related to this operator [34, 22]. Lemma 3.5 Let K E TH. There exists C > 0 such that forallv E [H1(K)j2. In addition to this, note the following lemma. Lemma 3.6 Let K E TH and suppose PK(v) has the property PK(v) nF = v nF. Then,forallv E [H1(K)j2 there holds: ii) = llviii) iiaMj((v)ilo,K:::; llvllo,K; IIPK(v)ilo,K :::; llvllo,K; 55
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iv) llvPK(v)llo,K :S Proof: See [2]. The following corollary holds as a result of the previous two lemmas: Corollary 3.7 There exist constants Cf' and such that i) IIPK(v)llo,K :S Cillvllo,K ii) llvPK(v)llo K :S forallv E [H1(K)]2. Proof: The two results clearly hold due to Lemma 3.6 when we assume PK(v) nF = v nF. Also, the two results hold by Lemma 3.5 by recalling PK(v) = 7rK(v) in the casepK(v)nF = IIF(vnF). 3.2 Consistency results We would like to establish results concerning the consistency of the methods. It turns out that the bulk of the work can be done for the symmetric general method, and the result for the general method will follow. We see that each of these methods is consistent. First, consider the result for the symmetric general method (2.50). Lemma3.8 Let (u,p) E H0(div,O) x [H2(0)n(0)] be the weak solution of (1.12) and ( u1 pi) be the solution from the symmetric general method (2.50). Then 56
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Proof: Under the assumptions of the lemma, we see that [p] IF = 0 on each internal edge F and RM (u,p, J)IK = 0 and Rc(u,g)IK = 0 for all K E TH. Using Remarks 2. 7 and 2.9, the facts RM ( u, p, f) = 0 and Rc ( u, g) = 0, and definition (2.49) of it follows, (w 1 ri)) = B((p(u),p), (p(wi), r 1)) 1 L aH2 (PK(u)P + TIK(p), PK(wi)r1 + TIK{r1))K KETH K L (V' uTIK(V' u), PK(wi)r1 + TIK(ri))K = B((uMu(RM),p), (p(wi), r1)) """ 1 M L...., aH2 (MP(R ), PK(wi)r 1 + TIK{r1))K KETH K + L (V' RcTIK(Rc), PK(wi)r1 + TIK(rd)K KETH 1 L (gTIK(g)aH2 (IKTIK)(Jx), PK(wi)ri)K KETH K = B((u,p), (p(wi), ri)) 1 L (gTIK(g)aH2 (IKTIK ){! x), PK(wi)ri)K KETH K 1 = {!, p(wi))nL (TIK(g) + aH2 (IKTIK ){! x), r1)K KETH K 1 L (gaH2 (IKTIK)(J x), PK(wi))K KETH K = (w1, ri)). The result follows directly from this. As a corollary to the last result, we may establish a consistency result for the general method (2.46). 57
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Lemma 3.9 Let (u,p) E Ho(div, n) X [H2(n) n L5(n)] be the weak solution of (1.12), (u1,p1 ) be the solution from the general method (2.46). Then Proof: Under the assumptions of the lemma, we see that IF = 0 on each internal edge F and RM(u,p, f)IK = 0 and Rc(u,g)IK = 0 for all K E TH. Therefore, by the proof of Lemma 3.8 we get B9((u,p), (w1 rt)) = (w1 rt)) = (!, p(wt))nL (nK(g) + (!.X+ c), rt)K KETH K 1 f (gaH2 (f x + CK ), PK(wt))K KETH K = B9((ut,Pt), (wt, rt)). The result follows directly from this. Remark 3.10 Another form of consistency exists. We recall problem (2.24): Find (3.8) the inclusions 58
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it follows that the exact solutions ( u, p) to problem to problem ( 1.12) satisfy B((u,p), (wH, TH )) = F(wH, rH ), from which it follows B((uuH,PPH), (wH, TH )) = 0, (3.9) we see B((uuH,PPH), (p(wi), ri)) = 0, (3.1 0) for all p(w1 ) E p(VI) and for all r1 E Qd. This result applies to (uH,PH) = ( u1 + + uf + PI + + pf + formed using solutions ( ui, PI) of the general method (2.46). Suppose that ( it1 p1 ) solve the symmetric general method (2.50). This method is derived by neglecting the term L + \lrdK KETH So, we may write that ( itH, PH) satisfies the modified form of (2.24): Find ( itH, PH) E (VI+ Ve) X (Qd + Qe) such that B((itH,PH ), (wH, TH )) L op(wt) + \lrt)K = F(wH, TH ), (3.11) KETH for all WH = p(wl) + Wb E p(VI) $ Vb and TH = r1 + Tb E Qd $ Wl.. Using the fact that B((u,p), (p(wt), rt)) = F(p(wt), rt)forall (wi, rt), we .find: B((uitH,PPH), (p(wt), rt)) =L ap(w1) + \lrt)K, (3.12) KETH for all WH = p(wt) E p(V1 ) and TH = ri E Qd. We conclude that the symmetric general method does not share the consistency result (3.10) with the general method. 59
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Remark 3.11 Notice the types of consistency results in Theorems 3.8 and 3.9 versus the consistency results of Remark 3.10 are qualitatively different from one another. The consistency results of the theorems indicate how the exact solutions with the relevant assumptions act inside the bilinear forms in the definitions of the numerical methods. This contrasts with the consistency results of the remark, which indicate how the numerical solutions act in the bilinear form definining the weakform of the Darcy problem. 3.3 Mass conservation property Notice that methods (2.46) and (2.50) are designed to ensure that uH E V1 + Ve satisfies (3.13) for all q1 E Qd. In the case that discontinuous pressure interpolation is used, we see that this implies in particular that (3.14) for each K E TH, and we see the methods are mass conservative. We document this in the following two lemmas: Lemma 3.12 Let ( u1 p1 ) be the solution to the general method (2.46) (respectively, (3.56)). Then, supposing the discontinuous pressure case, it holds, where uf is given by (2.37). 60
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Proof: This result follows directly from Equation (3.14 }, and the two facts that \7 = 0 and JK \7 = 0 (see (2.29) and (2.30)). Lemma 3.13 Let ( u1 p1 ) be the solution to the symmetric general method (2.50) (respectively, (3.58)). Then, supposing the discontinuous pressure case, it holds, i \7 (u1 = [g, where uf is given by (2.37). Proof: Same as the proof of Lemma 3.12. Remark 3.14 The above results are interesting. Although u1 is not itself locally mass conservative, we may update it with uf in order to recover this feature. We shall see that this property is useful in establishing convergence results as well. 3.4 Error estimates In this section, we show the methods are optimally convergent. In what follows, we define the errors: eu := up(ut), where (p(ut),p1 ) is the solution to the symmetric general method (2.50). We also define the interpolation error as: f/u := UVt, where v1 and q1 are interpolants which satisfy Equations (3.1) and (3.4), respectively. 61
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We first consider results for the symmetric general method (2.50) in Subsection (3.4.1). We find in Theorems 3.15 and 3.19 that (p(u1),p1 ) and (uH,PH). respec tively, converge in the meshdependent norm (2.52). Theorem 3.20 outlines the con vergence properties of p1 and PH in the L2norm, and Theorem 3.26 shows the con vergence of the locally mass conservative velocity of Section 3.3. As preparation to prove the results of Theorems 3.19 and 3.20, some results are developed in Theorems 3.21, 3.22, and 3.23. These last theorems will be referred to in other sections as well. Next, we present in Subsection 3.4.2 an analysis of the general method (2.46), with convergence results in the meshdependent norm presented in Theorems 3.27 and 3.28. Results for convergence in natural norms is given in Theorems 3.29 and 3.31. Finally, we will show in Subsection 3.4.3 that one term on the righthand side of the methods may be neglected in the general methods (2.46) and (2.50) due to its small impact on the error. 3.4.1 Estimates for the symmetric general method We begin by showing the convergence of (p( u1), p1 ) in the meshdependent norm (2.52). Theorem 3.15 Let (u,p) E ([H1(0)]2 n H0(div, 0)) x (H2(0) x (0)) be the solution of ( 1.12) and ( u1 pi) E V 1 x Q d be the solution of method (2.50). Then, there exists C > 0, independent of H, a, and a such that lll(up(ul),pPl)IIIH :S CH(v'I+"O) ( + )o:IPI2.n) Proof: Let v1 and q1 be interpolants of u and p, respectively. We have the following bound on the errors in the meshdependent norm, using its definition (2.52), the trian62
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gle inequality, and the interpolation results of Lemma 3.4 and part (ii) of Corollary 3.7, lll(eu, = lll("lu + Vtp(vt) + p(vt)p(ut), "lp + ql(3.15) Clll(p(vtUt)p(vt) + Vt, ql+IH77u, (3.16) C (lll(p(vtut), ql+ lll(vtp(vt), (3.17) Clll(p(vtUt), qt+ (3.18) So, we see the bound on the error in the meshdependent norm is given up to the term Ill (p( Vt ut), q1 pt) lllw We now seek a bound on this in terms of the interpolation error. To shorten the notation, define w1 := v1 u1 and r1 := q1 p1 Using the stability result of Lemma 2.15, the consistency result of Lemma 3.8, and the definition (2.49) of we have "lp), (wi, rt))l = I(TP("7u), p(wt))n + (\7 u, rt)u(\7 (wt), "lp)n 1 L H2 (PK("lu)"lp + llK("lp), PK(wt) + TtnK(rt))K KETH lT K + L FECH lT F 63
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We will now treat each of these terms separately. First, Young's inequality and the stability of p() (part (i) of Corollary 3.7) imply (ap(1Ju), p(wt))n 7 21 n + 2 1 n "fl C 7 21 n + 2 1 a liP( WI) n (3.19) "(I Next, integration by parts, identity ( 1.26), Young's inequality, and Lemma 3.2 imply (\7 ]u, rt)n =L (1Ju, \7ri)K + L (1Ju, [ri])F KETH FEt:H + 2 1 a L + L "(2 KETH FEt:H """ Q []2 + 2H C1"f II r1 llo,F FEt:H F 3 + 2 1 a L "(2 KETH + L (1 H K (; + H K 117uli,K) KETH K + 2Caa ( L + L ; "(3 KETH FEt:H F + 2 1 a L "(2 KETH + L a + Hkl11uli,K) KETH + 2Caa ( L + L ; (3.20) "( 3 KETH FEt:H F 64
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Next, (3.21) Next, Young's inequality, the triangle inequality, the fact that PK(ut) E to gether with the Poincare inequality, problem (2.1) (PK(ut) = \7PK(ut)), and the stability result for p( ) imply, (3.22) 65
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Finally, (3.23) Taking each of these inequalities together, we have when taking "Yl = "Y2 = "Y4 = "Y6 = "Y7 := 4ff, "'(5 := 2ff, and "'(3 := when o: 2: 1 and "YJ := 8ff when 0 < o: < 1. Applying the interpolation result in Lemma 3.4 and Equations (3.1) and (3.4) and collecting like terms, we find 2 2 (0' 2 1 12 ) lll(p(wi), r1)llln CH (1 + o:) ;luii,n + ;IP 2,n (3.25) 66
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Finally, we are able to determine from Equations (3.18) and (3.25) there exists C > 0 such that, (3.26) Remark 3.16 Notice again, as was remarked after the wellposedness of the symmetric general method was established, that there is in fact no dependence on o: in this method. As such, an implementation of the symmetric general method may assume o:=l. Remark 3.17 Notice that although the above theorem ensures the convergence of the solutions (p( ut), p1 ) asH is refined, this convergence has not been established in the natural norms. Rather; we have only shown, in particular; that '\1p1 converges to '\lp. The convergence ofp1 in the norm will be addressed momentarily. Remark 3.18 The theorem shows that we have achieved an optimal convergence of p( u1 ) to the true solution in the L2norm, based on the regularity assumption on the true velocity u. Convergence properties other than that established in the above theorem hold. In particular, we may show that the solution ( uH, PH) also converges to the exact solution in Ill(, )lllw Furthermore, we may show that PH and p1 convergerge at order H2 in the norm. These facts are stated in the next two theorems. The proofs will be given after some preliminary results have been established. 67
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Theorem 3.19 Let ( u1, P1) E V 1 x Q d be the solution of method (2.50) and assume the conditions of Theorem 3.22. Taking D M G PH = PI + Pe + Pe + Pe there exists C > 0, independent of H, o:, and a such that Proof: This is proved on page 76, after some preliminary results have been established. Theorem 3.20 Let ( u1, P1) E V 1 x Q d be the solution of method (2.50) and assume the conditions of Theorems 3.22 and 3.23. Taking D M G PH = P1 + Pe + Pe + Pe Furthermore, assume that f is constant throughout the domain. Then there exists C > 0 such that, liPPHIIo,n :S CH2 ( 1 + )a) (1 ( folult,n + IPI2.n), liPP1llo,n :S CH2 ( 1 + )a) (1 + ( folult,n + IPI2.n) Proof: This is proved on page 76, after some preliminary results have been established. The following three results (Lemma 3.21 and Theorems 3.22 and 3.23) must be established in preparation to prove the above two theorems. We first prove a lemma regarding the enrichment velocities and pressures. These results show, in particular, that certain quantities related to the enrichment will converge to zero. 68
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Lemma 3.21 Suppose (u,p) E ([H1(!1)]2 n H0(div, !1)) x (H2(!1) n is the solution of (1.12) and (v1 q1 ) E V1 x Qd are arbitrary. Several estimates regarding the enrichment velocities and pressures may be proved. 1. Let ( q:1 ) be solutions to problem (2.29). Then CTIIup(vl)llo,K + QI)IIo,K 2. Let be the velocity solution to problem (2.30). Then c HK [ 1 ( 1 )] live llo K luii.K + H llup(vl)llo K + QI)IIo K 1C' K 0" 3. Let vf be the velocity solution to problem (2.31 ). Then 4. Given Qe = qf + + q:1 there exists C > 0 such that I [ L C [(1 + a)lll(up(vi),p+ 4 KETH Proof: We begin with the first result. By definition, the fact that 1 1 (T (T and the triangle inequality, =II+ _!_ fV10" O,K 0" 0" O,K M 1 =live 0" O,K 1 = llup(vl) + qi)II 0" O,K 1 llup(vdllo K + QI)IIo K (3.27) (T 69
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We now consider the second result. By problem (2.30), integration by parts, the CauchySchwarz' inequality, the Poincare inequality using qCf E and the triangle inequality, it follows, 1 G G = (\1 qe 'V e ) K a 1 G G = (qe '\1 Ve )K a :S :S llK(Rc(v1,g))H 1 2 a a K O,K HK G ( c c 1 M ) :S llo,K IIR (v1,g)llK(R (vl,g))llo,K + aH'f< llqe llo,K :S (11Rc(v1, g)llK(Rc(v1, g))llo,K + II"Vq:1llo,K) :S (11\7 ullK(\1 u)llo,K + II"Vq:1llo,K) G ( 1 M ) :S CHKIIve llo,K II"V ullo,K + aHK II"Vqe llo,K (3.28) and so we conclude from the first result and the definition of lui1,K. K :S CHK [lull,K + H1 (11up(vl)llo K + .!_II"V(pQl)llo K)] (3.29) 7r K a which is the bound in the statement of the lemma. Next, consider the third result. Using problem (2.31 ), Corollary 2.13, and the proof of Lemma 3.1, we find, I I [ L = )a [ L KETH KETH I ::; c [ L (3.30) v FE[H a F 70
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So, the third result is established by noting = 0 under the assumptions of the theorem. Finally, we prove the final result. Using Lemma 2.13, the above two results, and problem (2.30), we have The following theorem establishes the error committed by ( v H, qH) is on the order of the error committed by (p( vt), qt) in the meshdependent norm. Theorem 3.22 Let (u,p) E ([H1(0)]2 n H0(div, 0)) x (H2(0) n (0)) be the solution of ( 1.12) and ( v 1 q1 ) E V 1 x Q d Furthermore, define Then, there exists C > 0, independent of H, a, and CT such that 71
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Proof: Using the definition of the norm, the triangle inequality, the trace inequality, and Lemma 3.21, we obtain 2 2 1"'"' 2 ili(uVH,PQH)iiiH = aiiuVHiio,u + IIV(pQH)iio,K KETH "'"' Q 2 + aH iillF([pQH])iio,F FECH F C [aiiu+ + +C L IIV(p+ L KETH KETH +C L (ii11F([p+ FECH C [lli(up(vt),p+ + +C L + L KETH FECH C [lli(up(vt),p+ + 1 "'"' 2 +C(1 IIVqelio,K KETH C(1 + a)2 (lli(up(vi),p+ Haiuii,u), (3.32) from which the conclusion of the theorem follows. It is useful to establish error estimates in natural norms. We will offer a proof on the convergence of PH and p1 following an approach based on Nitsche's trick. Assume (w1 rt) E V1 x Qd and consider first the following problem: Suppose 17 E L5(D) is the solution to the problem /::.17 = pTH in !1, v11. n = 0 on an, (3.33) where TH E Qd EB L5(TH ). Clearly, for a convex, Lipschitz domain n, 17 E H2(0) and there exists C such that i11b1 r H iio u With these facts in place, we are 72
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in a position to establish a result which expresses the error committed by functions r1 in terms of a consistency error and an error in the meshdependent norm Ill ( )Ill H' Theorem 3.23 Let (u,p) E ([H1(D)J2 n H0(div, D)) x (H1(D) n (!1)) be the solution of ( 1.12) and ( w1 r 1 ) E V 1 x Qd. Furthermore, let Assume 17 solves problem (3.33). We will suppose iJ E Vt n (!1) is the Clement interpolant (with the constant removed) of the solution 'f/, i.e., iJ = CH(17)IIn(CH(T/)) and r, E p(Vl) is defined by r, = \117) ). Supposing there is a positive constant cl such that H :::; cl H F for all interior edges F, there exists c > 0, independent of H, a, and a such that liPrHIIo,n :::; CH va ( 1 + )a) (1 +a) (lll(up(w1),pr1)IIIH + H valu!I,n) aB((uWH,PTH), (f!, r,)) + liPrHIIo,n liPrdlo,n:::; CHva ( 1 +)a) (1 +a) (lll(up(wi),pr1)IIIH + Hvalu!I,n) I +CHva(L: + aB((uwH,prH),(f!,r,)). KETH a liPrHIIo,n Furthermore, let where and be the pressure solutions of local problems (2.31 ), (2.29), (2.30), respectively. Then, liPrdlo,n:::; CHva (1 +)a) (1 +a) (lll(up(wl),pr1)IIIH + Hvalukn) aB((uWH,pTH), (f!, iJ)) + I liPTHI 0,!1 (3.34) 73
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Proof: It follows that '\lry E H(div, 0). Using the definition of the L2norm and continuity of the normal component of '\lry across edges gives us Now, add and subtract ('\lr,, '\l(prH ))n+ EFECH (1], [prH])F from equa tion (3.35), to obtain, 1 2 1 liPrnllon = ('\l(ryTJ), '\l(prn))n a a L (IFf!1]), {;f;;[prH]) FECH F + (vr,, '\l(prH )) L (1], [prH])F U FECH 1 ll'\l(ryTJ)IIonll'\l(prH)II a o,n I I + ff.(L HFII..!.'\lry7]112 )2 (2:: v ;FECH a O,F FECH a F L('T],[prn])F U FECH I +C fl HITJI2.n (L Ho: a o,n V FECH a F + L ('T],[prH])F !1 FECH CaH Va ( 1 +)a) liPrnllo,nlll(uUH,PrH)IIIn L('T],[prH])F (3.36) U FECH 74
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Now, observe that the definition ( 1.13) of B( ) and integration by parts yields, B((uwH,prH), (fJ, ij)) = ('Vij, _!_'V(prH))nL (fJ, [p(3.37) a FECH Substituting this and multiplying both sides of inequality (3.36) by the quantity 1 a 1 yields, prH o,n liPrHIIo,n::; CH v'a ( 1 +)a) lll(uUH,prH)IIIH aB((uWH,pTH), (fJ, ij)) + liPrHIIo,n Using Theorem 3.22, we find the first result. (3.38) From here, using reiK E L5(K) together with the Poincare inequality, the first result for r1 follows from the observation I liPTtllo,n::; liPrHIIo.n + ( L 2 KETH I ::; liPrHIIo,n + C ( L 2 KETH ( 1 2 ) 4 ::; liPrHIIo,n + CH v'a L KETH (3.39) Finally, under the assumption that r H = r1 + rf + + r:, part 4 of Lemma 3.21 implies ::; CH ( 1 +)a) (1 +a) (lll(up(wt),prt)IIIH + Hv'alult,n) B((uWH,PTH ), (fJ, ij)) liPrHIIo,n +CH 01{1 +a) (lll(up(wt),prt)IIIH + H v'alult,n) ::; CH (1 +)a) (1 +a) (lll(up(wt),pTt)IIIH + Hv'alult,n) aB((uWH,PTH), (fJ, ij)) + liPrHIIo,n 75 (3.40)
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and the final result of the theorem is proved. With the above three technical results established, we are in a position to provide a proof of the result of Theorem 3.19. Proof: [of Theorem 3.19] By Theorem 3.22, it follows that lll(uUH,PPH )IIIH::; C(l +a) (lli(up(ut),ppt)IIIH + H vaiuh.n) CH(l ( + )o:IPb.n). (3.41) Now, consider the proof of the result of Theorem 3.20. Proof: [of Theorem 3.20] Recalling equation (3.12) of Remark 3.10, we see B((uuH,PPH), (iJ, 17)) =L 'V17)K 1 ::; ( L 2 llafJ'V1711o,n (3.42) KETH Consider the term llaiJ'V1711o.n We find, using the triangle inequality, the defi nitions of r, and 17, along with the relevant interpolation results, that: ::; CHIIPPHIIo,n (3.43) Substituting this and the bound on )4 in Lemma 3.21 into equation (3.42), we find, aB((uUH,PPH), (iJ, 17))::; CHJl.7QIIi(up(ul),pPdiiiHIIPPHilo,n 76
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Now, the relevant result of Theorem 3.23, and collecting like terms, implies: liPPHIIo,n CHy'a ( 1 + Ja) (1 +a) (lll(up(ut),ppt)IIIH + Hvalult.n) Finally, observing, we get the first result of the theorem: (3.44) The second result follows similarly. Remark 3.24 As has been remarked before, the value of a is unimportant both for the wel/posedness and the convergence of the symmetric method, and so we may assume a= 1. Remark 3.25 In particular, Theorem 3.20 establishes the optimal convergence of p1 in the L2norm. We close this section with a result on convergence in the divergence norm and seminorm when discontinuous pressure interpolation is assumed. Lemma 3.26 Let (u,p) E ([H2(0)fnH0(div, 0)) x (H2(0)n(0)) be the solution of ( 1.12) and ( u1 p1 ) E V 1 x Q _1 be the solution of method (2.50). Given from equation (2.31 ), it follows that there exists C > 0 such that, ll\7 (uUt CHiul2.n, lup(u,)CH [(1 + n) ( Jluh.n + + lub,n] 77
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Proof: First, it follows from Lemma 3.13 that for arbitrary K E Tn, (3.45) Therefore, we have the following by applying the Poincare inequality and the fact that "\1 ( Ut + IK is constant: II"V u"V (ut + = II"V u9"V (ut + + 9llo,K :::; CHKII"V"V (uUt:S C H K II"V "\1 ullo,K :S CHKiui2,K From here, the first result follows by observing, II"V (uUt= L II"V (uUt< C L KETH We now prove the second result using Theorem 3.15 and Lemma 3.21. llup(ut):S llup(ut)llo,n + :S p(ut),pPt)llln + ya (3.46) (3.47) :SCH(l+a)( /IIultn+_!_IPbn) (3.48) y; a Recalling the definition and "V p(u1 ) = "V Ut. llup(ut)= llup(ut)+ II"V (uUtand applying inequality (3.48) along with the first result of the theorem, we find the second result of the theorem. 78
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3.4.2 Estimates for the general method The approach to the analysis will mimic the development of the previous subsec tion. First, we estimate the error committed in the meshdependent norm. Theorem3.27 Let(u,p) E ([H1(n)FnH0(div,n)) x lution of ( 1.12), ( u1 p1 ) E V 1 x Q d be the solution of method (2.46), and (p( ui), pi) be the solution of method (2.50). Assuming the conditions of Lemma 2.17, there exists C > 0, independent of H, a, and a such that lll(up(ui),ppi)IIIH :S CH (1 +a) ( + )a:IPI2.n). Proof: Notice that and PPI = PPI+ PIPI := ep + TJ. With this, we may write 79
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Theorem 3.15 gives us a bound on the first term on the righthand side. We must estimate the second term. Using Lemmas 2.17, 3.8, and 3.9, !3111(wt, :S B9((wt, rt), (wi, rt)) = B9((p(u1)u),p1p), (wt, rt)) +B9((up(iit),piit), (wt, rt)) = B!((up(ui)),pPt), (wt, rt)) +B9((up(iit),ppi), (wt, rt)) + L Pt]), ap(wi) + \7ri)K So, we see lll(wt, :S lll(eu, Combining this with (3.49) we see (3.51) So, we have established the theorem. As was done for the symmetric method, we may use Theorems 3.22 and 3.23 to establish the following two results. 80
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Theorem 3.28 Let (ii.1,p1) E V1 x Qd be the solution of method (2.46) and assume the conditions ofTheorems 3.22 and 3.27. Taking D M G PH = Pt + Pe + Pe + Pe there exists C > 0, independent of H, a, and a such that Proof: The proof is exactly the same as for Theorem 3.19. Theorem 3.29 Let (ii.1,p1 ) E V1 x Qd be the solution of method (2.46) and assume the conditions of Theorems 3.22, 3.23, and 3.27 and take D M G PH = Pt + Pe + Pe + Pe Furthermore, assume that f is constant throughout the domain. Then there exists C > 0 such that, liPPHIIo,n CH2 ( 1 + )a) (1 + a)2 ( folult,n + IPI2.n) liPPtllo,n CH2 ( 1 + )a) (1 + a)2 ( foluh,n + IPI2,n) Proof: We first observe that equation (3.1 0) of Remark 3.10 tells us for this solution ( u H, pH), we have The result then follows the steps taken in the proof of Theorem 3.20, keeping this simplifying result in mind. 81
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Remark 3.30 Although we have shown these results for p1 E Qd, it is important to note that if in fact p1 is continuous, then the term which makes this formulation nonsymmetric disappears and the formulation is symmetric. As such, we do not see any dependence on a in this method when considering continuous pressure interpolation. Finally, we may establish a type of convergence in the divergence seminorm. Lemma3.31 Let(u,p) E ([H2(n)FnH0(div,n))x(H2(f2)nL5(n))bethesolution of (1.12) and (iit,Pd E vl X Qd be the solution of method (2.46). Given uf, it follows that there exists C > 0 such that, Proof: The result follows using Lemma 3.12 and the steps presented in the proof of Lemma 3.26. 3.4.3 Results for a general g E H1(0). Recall the definition of given by problem (2.30). Define := the portion of which relies on the righthand side 9IK IIK(g). Next, consider the term on the righthand side of the methods (see, e.g., (2.48)). L (g, PK(vi))K. (3.52) KETH We have, using PK(v1 ) E L5(K), integration by parts, n = 0, and problem (2.1), (3.53) 82
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We also see that (3.54) The proof for this is given in [2], but it will be presented here for completeness. Observing = and applying CauchySchwarz' inequality, (3.55) Since this term appears to be small, we consider methods in which this term is neglected. First, the general finite element method becomes: Find (u1,pt) E V1 x Qd such that (3.56) for all ( v 1 q 1 ) E V 1 x Qd, where B9 is defined in equation (2.47) and qt) := {!, p(vd)nL (nK(g) + a;2 (IIlK){! x), q1) KETH K I\ + L f x, PK(vt)) (3.57) KETH K K Also, the symmetric method will read:Find (u1,p1 ) E V 1 x Qd such that (3.58) 83
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for all (vi, QI) E VI x Qd. We have taken the definition of from (2.49). In light of the preceding discussion, consider the following lemma. Lemma 3.32 Assume ( ui, PI) is the solution to method (2.46) and (it. I, PI) is the solution to method (3.56), and let g E HI (0). Then, (3.59) Proof: By the first Strang Lemma (see, e.g., [22]) it holds: lll(p(uiit.I),PI PI)IIIH:::; sup (p(v, )m)Ep(V,) xQr {0} L:KETH ( O'pK( VI) )K Ill (p( VI), QI) IIIH 1 :::; C [ L KETH I :::; CH [ L 119KETH :::; CH2Ig!I,u. (3.60) A similar argument holds when the symmetric methods are considered. Lemma 3.33 Assume (u1,pi) is the solution to method (2.50) and (it1,pi) is the solution to method (3.58), and let g E H1 (0). Then, (3.61) Remark 3.34 Lemnuis 3.32 and 3.33 indicate to us that the inconsistent methods (3.56) and (3.58) have the same performance as (2.46) and (2.50), respectively, since the consistency error will be at order H2 As such, the term will not be computed in each of the numerical tests we perform in Chapter 5. However, note that the term 84
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is easily computed should one care to include the term when calculating the right hand side of the method, and so the term may be excluded simply because it does not undermine the convergence of the method. 3.5 Concluding remarks This chapter has been devoted to an analysis of methods with lefthand sides ) and BY(,) under the assumption that the exact function p(ut) is known. Implicitly, this proves a convergence for four separate methods since PK(u1 ) takes on one of two different boundary conditions; recall, either (3.62) or, (3.63) We have seen that p(u1 ) and p1 as well as uH and PH converge at optimal rates in the 2norm. Furthermore, we have found a mass conservative velocity u1 + uf which converges in the H(div, !1)norm. As has been noted repeatedly, whenever p(ut) satisfies boundary condition (3.62), we know PK(ut) = rrK(ut) and the meth ods become (2.68) or (2.65), respectively. Since these methods have terms which may be computed exactly, they automatically inherit the analysis of this chapter. However, we don't know the exact values of PK(u1 ) and PK(ut) when p(ut) satisfies boundary condition (3.63), and so we expect the approximation assumed in defining methods (2.74) and (2.72) should introduce a perturbation to the theory presented in this chap ter. In the following chapter, we take the step of defining the approximation using a second level. We will then show that this approach, under appropriate assumptions, does not affect the convergence rates we have found in this chapter. 85
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4. Error Analysis of the Fully Discrete Methods In Chapter 3, we established, under the assumption that we know the exact solu tions to problem (2.29) (for both choices of boundary condition), the methods have a unique solution which will converge under the correct assumptions on the true so lution. However, we only know the solutions to problem (2.29) for one choice of boundary condition, whereas the exact solution is unknown for the other choice of boundary condition. In this chapter, we firrnly establish the choices made for the value of the approximate local solutions in the methods (2.72) and (2.74). We seek this local approximate solution by solving a local finite element method on each tri angle K E TH. and as such, we establish methods (2.72) and (2.74) as twolevel finite element methods. Note that since the results for methods (2.68) and (2.65) are established, we assume throughout this chapter that PK(wi) and PK(wi) satisfy the following problem for w 1 E V 1 (4.1) The outline of the chapter is as follows: Following the development in [2], we first establish our approach to approxi mating the solutions of (4.1) by solving a Laplacetype problem with a local finite element method in Section 4.1. With the approximate solution in place, we give an explicit definition of the fully discrete methods (2.72) and (2.74). 86
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In Section 4.2, we present results related to a numerical analysis of the fully discrete methods. In particular, we see the methods are wellposed, they have a small consistency error, and have convergence properties similar to those for the general methods (2.46) and (2.50) as presented in Chapter 3. 4.1 The Fully Discrete Methods In order to make explicit the exact form of the approximate operators introduced in methods (2.72) and (2.74), we adopt the approach considered in [2]. First, recall that Remark 2.7 implies (4.2) Furthermore, for this particular PK we have established in Section 2.6.2 (4.3) We substitute this into both the symmetric and full methods ((2.50) and (2.46), re spectively). First, with this substitution, the symmetric general method becomes: Find (u1,pt) E V1 x Qd such that (4.4) 87
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(vi,QI)) := L PI+ IIK(PI), QI + IIK(qi))K KETH a K L (IIF{[pi]), IIF([qi]))F, (4.5) FEE:H a F F9(vi, QI) := (/, vi)nL (rrK(g) + a;2 (IKIIK )(! x), QI) KETH K K + L (ag;2 f (4.6) KETH K K Likewise, the general method becomes: Find ( ui, pi) E VI x QI such that B9((ui,pi), (vi, QI)) := B((\7 (\7 qi)) + L (V}(([pi]), a\7 + \lqi)K (4.7) L PI+ IIK(PI), QI + IIK(qt))K KETH K L a: (IIF([pi]), IIF([qi]))F FEE:H (4.8) From equation (4.2) and problem (4.1) it follows that the function E solves the strong local problem = \7 vi, inK, (4.9) n =VI n, on 8K. 88
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This implies a weak formulation for the problem (4.9): Find E H1(K) n such that (4.10) for E H1(K) n We seek to build an approximation to using this weak form of the local problem. Following [2], let {T,.K} be a family of regular partitions of K E TH into triangles [< and define K 0 2 K 'Rh := C (K) n L0(K) : E IP't(K), 'VK E T,. }, (4.11) where IP'1(K) stands for polynomials up to order l on I<. Let h =max{ diamf< : [< E 'li.}. We propose an approximate solution of (4.9) through the following standard Galerkin method: Find Ph ( Vt) E 'Rf': such that (4.12) for all E 'Rf':. Remark 4.1 We see that Ph(v1 ) is in fact the projection of into the space Rf': defined by the following Galerkin orthogonality condition: (4.13) for all E 'Rf':. Taking the solution Ph(v1 ) E Rf defined by solving (4.12), we may substitute into problems (4.4) and (4.7). In particular, we first see the fully discrete symmetric method: Find (ut,h,Pt,h) E Vt x Qd such that: (4.14) 89
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where Bs,h((vl, Q1), (w1, r1)) := L (aVph(vl), Vph(wi))K KETH (V v1,r1)n(V wl,QI)n 1 L aH2 (aph(vl)Q1 + IIK(QI), aph(wl)r 1 + IIK(r 1))K KETH K (4.15) Fh(w1, r1) := (/, wt)12L (rrK(g) + ((IKIIK )(! x), r1) KETH K K + L (ag;2 f x,ph(wt)) (4.16) KETH K K Likewise, the fully discrete full method is: Find ( u1,h, p1,h) E V 1 x Qd such that: (4.17) where Bh((vl, Q1), (w1, r1)) := L (aVph(vt), Vph(wi))K KETH (4.18) Remark 4.2 Comparing method (4.14) (or (4.11)) to method (2.74) (resp., (2.12)), we see they are the same when taking PK(wt) .Vph(wl) and PK(wl) := 90
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We have now defined methods (2.72) and (2.74) in tenns of (4.17) and (4.14), respectively, and we are ready to provide a numerical analysis of them. 4.2 Numerical analysis ofthe fully discrete methods. The goal of this section is to provide an analysis of the fully discrete methods (4.14) and (4.17). Before proceeding to the analysis, we consider some preliminary results in Subsections 4.2.1 and 4.2.2. We then tum to the analysis, with which we es tablish existence and uniqueness results (Subsection 4.2.3), consistency results (Sub section 4.2.4), and convergence results (Subsection 4.2.5). 4.2.1 A relationship between two velocities. Recall the result (stated in Lemma 3.6, proved in [2]) that for v1 E V t. the operator p( v t) is stable in the sense that there exists C such that ( 4.19) As such, we conclude that p(vt) = 0 whenever v1 = 0. As a corollary to Lemma A.6, we have the following, which shows the converse inequality holds as well. Lemma 4.3 There exist positive constants C, C1 andC2 independent of H such that,forall (vt,Qt) E V1 X Qd CIIPK(vt)llo,K :S llvtllo,K :S CdpK(vt)llo,K lll(vt, qt)lll" :S C2lll(p(vt), qt)lllw (4.20) (4.21) Proof: The first inequality of (4.20) is a restatement of part (i) of Corollary 3.7. We may establish the second inequality of ( 4.20) by observing the existence of C > 0 91
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such that I llv1llo,K :S CHIIIvl nllo,aK I = CHIIIPK(vi) nllo,aK :S C + HiciPK(vt)li,K :S CIIPK(vi)IIo,K (4.22) The last inequality above follows from the inverse inequality, where we used PK(Vt) is a finite element space (see Lemma A.6). Next, (4.21) follows directly from (4.20) and the definition of Ill(, )lllw Lemma 4.4 There exists C such that Proof: See [2]. IPh(v)II,K :S llvllo K IMi<(v)Ph(v)b,K :S Chlvh,K, Next, we define the meshdependent norm (4.23) (4.24) This norm contains terms driven by the terms present in the fully discrete methods ( 4.14) and ( 4.17). As such, it represents the most direct norm in which to prove wellposedness for these methods. Nevertheless, we would like to prove convergence results for the methods in a norm not containing the discrete operator 'Vph( vi). The following lemma clarifies the conditions under which the norm Ill(, )lllh is equivalent to the norm Ill(, )IIIH defined in (2.52). 92
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Lemma 4.5 There exists a constant C0 such that if h :$ C0H K for all K E Tn, then there exists a constant a positive constant C such that (4.26) Proof: First, we have ll\i'ph(viHo,K :$ llvdlo,K by Lemma 4.4. So, (4.27) Next, according to item (i) of Lemma 4.3 and Equation (4.2), the definitions of 111(, )llln and Ill(, )lllh' the triangle inequality, Lemma 4.4, and the inverse inequality we have the following lll(vi, :$ Clll(p(vt), = C( L + L KETH KETH + L FECH F :$ C ( L Ph(vdli,K + lll(vi, KETH :$ C (a L h2lvdi,K + lll(vi. KETH :$ C (a L h2 + lll(vi. KETH Making the assumption h :$ Jfc. we have (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) Subtracting from both sides, we see the result holds with C0 = Jzc. 93
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Remark 4.6 Much like Lemma 4.4, this last result implies that if we are able to show that the there is a unique solution Ph(v1 ) which solves either problem (4.14) or (4.17), then there is a unique v1 for which this is true. In fact, this is the approach we will take in showing there is a unique solution (v1,q1 ) E V1 x Qd which solves these problems. 4.2.2 Approximate enrichment solutions It turns out that some approximations to the enrichment solutions must be made in accordance with the fact that we must approximate \7 u1,h). Recalling problem (2.30), we know that the enrichment velocity should satisfy, (4.35) We have also remarked that (4.36) Our inability to compute exactly means we are unable to compute ( and therefore ( exactly. Instead, we must first replace ( with approximations defined by, := aph(Ut,h)+ (IKIlK)(! XPI,h)iK, (4.37) where Ph ( u 1,h) is the approximation ( u 1,h) defined by the local finite element problem (4.12). 94
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We next replace ( with ( which are defined according to the problem (modified from (2030}}, + = 0, inK = Rc(ut,h,g)IIK(Rc(uL,h,g))Hl 2 inK, a K on F c 8K 0 (4o38) Then, denoting by ( the solutions of (2o3l) (with p 1,h used in the righthand side of the equation}, we define the (approximate) enriched solution ( uH,h, PH,h) by, (4o39) 0 M D G PH,h oPl,h + Pe,h + Pe,h + Pe,ho (4.40) Based upon these definitions, it would be nice to know explicitly what weak problem is satisfied by the enriched solution ( uH,h, PH,h)o Recall identity (2o35)o As we did in establishing that identity, we may use integration by parts and the identity ( lo26) to establish, L IIF([rtD))Fo(4.41) FEt:H We also note that an identity similar to (2o34) may be established, again when inte grating by parts and applying the boundary condition 0 n = 0, L KETH = L ('\lo aph(wt)Tt + IIK(rL))Ko KETH 95 (4.42)
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Using identities ( 4.41) and ( 4.42) in the fully discrete full method ( 4.17), collecting velocity terms according to equation (4.39), and observing that + pf,h + pf,h, \7 w 1 ) = 0, we see that the enriched solution ( uH,h, PH,h) arising when solving the fully discrete full method ( 4. 17) satisfies, L (auH,h, \i'ph(wi))K(\7 uH,h, r1)n(\7 Vt,PH,h)n = F(w1, r1), (4.43) KETH We need to establish some estimates on the approximate enrichment solutions. These estimates are important in establishing certain convergence properties in the sequel. Note that since (uf,h,pf,h) are explicitly defined in terms of (u1,h,PI,h) via equations (2.37) and (2.38), the estimate established in Lemma 3.21 holds. Bearing this in mind, consider the following lemma: Lemma 4.7 Suppose (u,p) E ([H1(n)J2 n H0(div, n)) x (H2(n) n L6(n)) is the solution of ( 1.12) and ( v 1 q1 ) E V 1 x Q d are arbitrary. The following estimates on the approximate enrichment velocities and pressures hold: 1. Let be defined as in equation (4.37). Then 2. Suppose (uf,h,pf,h) solves problem (4.38). Then lluf.hllo,K :S C ( HKiuii,K + lluPK(ui,h)llo.K + llu1 +;:IIV'(pPI,h)llo,K 96
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3. Suppose Pe,h := + + where is given by (2.38) (using Pl,h). Then I [ L :S CJ1 + alll(uU1,h,pPl,h)IIIH KETH +Ji711up(ul,h)llo,n + Ji1Hiul1.n Proof: We begin with the first result. First, recall from (4.37) the definition of the approximate enrichment pressure Next, from (2.41), along with the observation in Remark 2.7, the form of the exact enrichment pressure based upon the solutions ( u1,h, p1,h) is, From here, repeated application of the triangle inequality, part l of Lemma 3.21, the observation VMi<(w) = PK(w), the fact that h:::; HK. and Corollary 3.7 imply 1M 1M M 1M 11VPe,hll :S 11V(pe Pe,h)ll + 11VPe II a O,K a O,K a O,K = IMi<(ul,h)Ph(ul,h)h.K + a O,K :S lluV Mi<(ul,h)llo,K + lluVph(ul,h)llo,K :S lluV Mi<(ul,h)llo,K + lluVph(u)llo,K + IIVPh(uul,h)ll 1 a O,K :S lluVMi<(ul,h)llo,K + lluVMi<(u)llo,K + +IIVPh(uul,h)ll + IIVMi<(u)Vph(u)llo,K :S C (lluPK(ui,h)llo,K + HKiuh,K + lluui,hllo,K) 1 +CIIV(pPl,h)llo K' (4.44) a 97
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which is the first result. Now, consider the second result. We inherit equation (3.28) from Lemma 3.21, which tells us (4.45) From here, we apply the first result of this lemma to find which is the second result. Finally, the last result follows by the triangle inequality, parts I and 2 of the present theorem, Lemma 2.13, and the definition of Ill(, )IIIH ""' 1 2 ""' (1 D 2 1 M 2 1 G 2 ) ;:IIY'Pe,hllo,K :S C ;:IIY'Pe,hllo,K + ;:IIY'Pe,hllo,K + ;:IIY'Pe,hllo,K KETH KETH :S C [a L + L <711uFElH KETH +C L (<7Hkluli.K + <711u+ KETu :S C(1 + a)lll(uui,h,P(4.47) from which the result follows. 4.2.3 Wellposedness results We first show that each of the twolevel methods considered in this chapter are wellposed. 98
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4.2.3.1 Wellposedness of the fully discrete symmetric method. We will proceed with the same approach adopted to prove the stability of the general methods. Lemma 4.8 Let B8,h be the bilinear form defined in (4.15). Under the conditions of Lemma 4.5, it follows that there exists f3 independent of H, h, and a such that (4.48) and we conclude that, sup BB,h((vt, Qt), (wt, rt)) 2: fJIII(vt, qt)IIIH, (4.49) (w.,rl)EVI xQr{O} Ill( Wt' Tt)IIIH forall (vt,Qt) E V1 x Qd. Proof: By the definition of BB,h we have 99
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So, it follows that (4.51) when we take C = min { J1 } Now, by Lemma 4.5 it follows that (4.52) where f3 = .g, where Cis the constant in Lemma 4.5. Finally, the second result of the lemma follows by observing that lll(vt, qt)IIIH = lll(vl, qt)lllw 4.2.3.2 Wellposedness of the fully discrete full method We now establish the fully discrete full method has a unique solution. Lemma 4.9 Let Bh be the bilinear form defined in (4.18). Under the conditions of Lemma 4.5 and the assumption a ::; 2C4(l+CJ)' it follows that there exists f3 indepen dent of H, h, and a such that (4.53) and we conclude that Proof: First, observe that we may write L \lqt)K. (4.55) KETH 100
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We may write Using this and Equation (2.59) of Lemma 2.17 in Equation (4.55), we find there are positive constants "Yl and 72 such that where C = min{4, 2b) The first result of the lemma follows now from using Lemma 4.5 to write (4.57) 101
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where /3 = b where Cis the constant in Lemma 4.5. Finally, we may conclude the second result of the lemma follows by observing that Ill ( v1, qt) Ill H = Ill ( v1, q1) lllw 4.2.4 Consistency results Due to the fact that we have replaced PK(v) with an approximation, the consistency of the general methods is not inherited by the fully discrete methods (4.14) and ( 4.17). The following theorem measures the error introduced into the methods via this approximation. For this subsection, define eu := u u 1,h, ep := p P1,h where (u1,h,P1,h) solves either method (4.14) or method (4.17), as determined by the situation. First, consider the consistency result for the fully discrete symmetric method. Lemma 4.10 Let (u,p) E {[H1{0)]2 n H0(div, 0)) x {H2(0) n (0)) be the weak solution of ( 1.12) and ( u1,h, P1.h) be the solution of (4.14). Then, 2a1 [ 2 2 J C I 2 Bs,h((eu, ep), (w1, rt)) :S Ch 2 lult,n + ll9llo,n + 2111 (w1, r1)IIIH, for all ( w 1 r 1 ) E V 1 x Q d when 1 is a positive constant. Proof: First, under the assumptions of this lemma, we have = 0 over internal edges F E &H. Observe next, using the definition (4.15) of Bs,h(, ), PK(u) and definition (2.49) ), = Bs,h((u,p), (w1, rt))Fh(w1, rt) = L (aVph(u), Vph(wt))K(V u, rt)u(V w1,p)n KETH 1 L aH2 ( aph(u)p + IIK(p), aph(wi)r1 + IIK(ri))K KETH K Fh(w1, r1) 102
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= (ap(u), p(wi))n(V u, Tt)n(V Wt,P)nFh(wt, Tt) + L [(aVph(u), Vph(wi))K(apK(u), PK(wt))K] KETH 1 L H2 (PK(u)P + IIK(p), PK(wt)Tt + IIK(rt))K KETH 0" K 1 + L aH2 (PK(u)P + IIK(p), PK(wt)Tt + IIK(ri))K KETH K 1 L aH2 ( aph(u)P + IIK(p), aph(wt)r1 + IIK(r1))K KETH K = (wt, ri))Fh(wt, Tt) + L [(aVph(u), Vph(wt))K(apK(u), PK(wt))K] KETH + L ( aMi<(u)p + IIK(p), aMi<(wt)r1 + IIK(r1))K KETH K 1 L aH2 ( aph(u)p + IIK(p), aph(wt)r1 + IIK(ri))K KETH K = F9(Wt, Tt)Fh(Wt, Tt) + L [(aVph(u), Vph(wi)PK(wt))K + (a(Vph(u)PK(u)), PK(wi))K] KETH + L (a(ph(u)Mi<(u)), aph(wt)Tt + IIK(ri))K KETH K + L ( Mi<(u)p + IIK(p), a(ph(wt)Mi<(wt)))K. KETH K We will bound these five tenns. First, the Galerkin orthogonality ( 4.13) and Ph ( u) E together with PK(w 1 ) = VMi<(wt), imply (4.59) 103 (4.58)
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Next, Young's inequality, item ( ii) of Lemma 4.4, and the stability of p( 0 ) imply (a(V'ph(u)p(u)), p(wi))n :S L IIV'Ph(u)+; KETH "f :S C L IPh(u)+; KETH "f :S C [ 'Y; h2luli,n + 0 (4) Also, using Ph(u) E along with the Poincare inequality and item (ii) of Lemma 4.4, we obtain, and so we find (4061) Also, 104
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= L a(ph(wi)K KETH K L a(ph(wi)KETH K = L (4.62) KETH a K Finally, using the definitions (2.48) and (4.16) of F9(w 1,ri) and Fh(w1,r1), respectively, we have = (/, wi)nL (ITK(g) + a;2 (IKITK )(! x), r1)K KETH K + L (ag(IKIlK)(/ KETH K (/, w1)nL (ITK(9) + a;2 (IKITK )(f x), ri)K KETH K 1 + L (agH2 (IKITK )(! x),ph(w 1))K KETH K =L (ga(ph(wi)(4.63) KETH K Taking all of these inequalities together, 105
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C '; h2JuJi,nL (M'f<(f), a(ph(wl)M'f<(wt)))K KETH K + lll(wh L (gM'f<(f), a(ph(wl)M'f<(wi)))K 1 KETH a K L (g, a(M'f<(wl)Ph(wt)))K The result follows by rearranging. We now consider the consistency result for the fully discrete full method. Lemma 4.11 Let (u,p) E ([HJ(O}F n H0(div, 0)) x (H2(0) n be the weak solution of ( 1.12), ( u1,h, p1,h) be the solution of (4.14). Then 106
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Proof: First, under the assumptions of the theorem, we have [p]IF = 0 for each internal edge F. Therefore, it follows that Bh((eu, ep), (w1, r1)) = Bs,h((eu, ep), (w1, rl)) :S Ch2a2 1 + + (4.64) Remark 4.12 Equation (4.43) is useful because "consistency" results similar to those in Remark 3.10 may be established. Noting that for the p( ) considered in this chapter, (4.65) it follows that the enriched solution ( uH,h, PH,h) arising when solving the fully discrete full method ( 4.17) have the property, (4.66) for all ( w 1 r 1 ) E V 1 x Qd. This should be compared to the consistency result (3.1 0) of Remark 3.1 0. Moreover, define { Ue,h, Pe,h). where , M G D (4 67) PH,h := Pl,h + Pe,h + Pe,h + Pe,h to be the enriched solution arising when solving the fully discrete symmetric method (4.14). We see this satisfies (4.68) 107
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for all ( w 1, r1) E V 1 x Q d This follows by observing that the symmetric method is de rived by removing the tenn LKETH a\1ph(wt) + \7r1 ) Kfrom the full method. We are then able to determine that: B((u,p), (p(wi), r1)L (ait.H,h, \1ph(wt))K + (\1 it.H,h, rt)n + (\1 Vt,PH,h)n KETH =L a\1ph(w1) + \1r1) K. (4.69) KETH for all ( w 1 r 1 ) E V 1 x Q d This should be compared to the consistency result (3.12) of Remark 3. 10. 4.2.5 Convergence results. With the existence of a unique solution established, as well as an understanding of the consistency error related to the fully discrete methods (4.14) and (4.17), we are in a position to establish convergence results for these methods. We will work first with the fully discrete symmetric method ( 4.14 ), and close with results related to the fully discrete full method. In much the same fashion as we did with the general methods in the analysis of Chapter 3, we will establish three main results: convergence in meshdependent norm, convergence of pressure in the 2norm, and convergence of velocity in the divergence norm. 4.2.5.1 Convergence results for the symmetric fully discrete method. We are now in a position to show that the solutions obtained by solving the fully discrete symmetric method will converge to the true solution under the correct assumptions on the true solution. 108
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Theorem4.13 Let (u,p) E ([H1(0)]2 n H0(div,O)) x (H2(0) n be the solution of (1.12) and (u1,h, Pt,h) E V1 x Qd be the solution of method (4.14). Under the assumptions of Lemma 4.5, there exists positive constant C such that lll(uUt,h,PPt,h)IIIH::; c [(H + h)(l + + H(l + va) )a1PI2,n] +Chv'aiiYIIo,n Proof: Let v1 and q1 be interpolants of u and p, respectively, which satisfy the interpolation properties (3.1) and (3.4), respectively. Define eu := u Ut.h ep := pPt,h "lu := uVt. and "lp := pq1 In addition, we define w 1 := v1 u1,h and r1 := q1 Pt,h We have the following bound on the errors in the energy norm using its definition, the triangle inequality, and the interpolation result of Lemma 3.4, Ill ( eu, ep) = Ill ( "lu + Wt, "lp + TJ) ::; C [lll(wt. + lll("lu, ::; C [111(wt, + CH2(alultn + 1 : a (4.70) So, we see the bound on the error in the energy norm is given up to the term Ill ( Wt, r1) lllw We now seek a bound on this in terms of the interpolation error. Using stability, consistency, and the definition ofB8,h(, ),we have tJIII(wt. BB,h((wt, rt), (wt, rt)) = BB,h((eu, ep), (wt, rt))BB,h((ryu, "lp), (wt, rt)). (4.71) 109
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Now, observe 1 L aH2 ( aph(TJu)7]p + IIK(TJp), aph(w1) + r1IIK(r1))K KETH K + L (4.72) FECH a F We will now treat each of these terms separately. First, Young's inequality and Lemma 4.4 imply, L (a\lph(TJu), \lph(w1))K + KETH /1 C + (4.73) Next, we inherit the bound given in (3.20), As for the third term, consider the bound given by (3.21 ), 110
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Moreover, Young's inequality, the triangle inequality, the Poincare inequality applied to Ph(u1 ) E and Lemma 4.4 imply, (4.76) Finally, (4.77) Ill
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Taking each of these inequalities together, we have when taking ')'1 = ')'2 = ')'4 = /'6 = /'7 := 4f, 'Ys := 2f, and ')'3 := when a 2: 1 and 1'3 = 8f when 0 < a < 1. Applying the interpolation result in Lemma 3.4, Equations (3.1) and (3.4), and collecting like terms, we find lll(wt, :S CH2 (1 +a) n + n) + B8,h((eu, ep), (wt, rt)). a a (4.79) Using the consistency result in Lemma 4.10 with ')' = 2f, we then see lll(wt, :S C [(H2 + h2)(1 + + H2(1 + + (4.80) 112
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Finally, upon substituting this result into equation (4.70), we obtain, lll(eu, :S C [(H2 + h2)(1 + a);luli,u + H2(1 + + (4.81) and the result follows. The following theorem establishes that the convergence of (p( wi), rJ) will be controlled by the convergence of (w1 r1). This result will help us show the rate of convergence of the approximate pressure PI,h in the L2norm. Lemma4.14 Let(u,p) E of ( 1.12) and ( w1 r1 ) be functions in V 1 x Qd. Then there is a constant C such that llup(wi)IIo,u :S C ( Hluh,u + lluwdo.u), lll(up(wi),prJ)IIIH :S C(lll(uw1,pri)IIIH + Hvaluh,u). Proof: First, using the triangle inequality and Lemma 3.7, we have llu= L llu:S C L (ilu+ llp(uKETH :S C L (Hi
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As such, we conclude lli(up(wi),p= aiiu+ L IIV(pKETH + L lillF([pFECH F :SaC + + L IIV(pKETH The result of the theorem follows from this. As a consequence of the above lemma, we have the following error estimates for p( u1 ,h) We first establish convergence in the energy norm, then we consider the convergence of the pressure p1 ,h in the norm. Theorem 4.15 Let (u,p) E ([H1(n)J2 n H0(div, n)) x (H2(n) n L5(n)) be the solution of (1.12) and (u1,h, p1,h) E V1 x Qd be the solution of method (4.l4).Under the assumptions ofTheorem 4./3, there exists C > 0 such that lli(up(ul,h),pPl,h)IIIH :S C [(H + h)(l + + H(l + vfa) )a:1PI2,n] +Chvall9llo,n Proof: The theorem follows by a direct application of the result of Theorem 4.13 to the result of Lemma 4.14 when we choose w1 = u1,h and r1 = Pl,h We now state and prove the theorem that, in particular, p1,h will converge at order H2 in the 2norm. 114
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Theorem 4.16 Let (u,p) E ([H1(0)]2 n H 0 (div, 0)) x (H2(0) n (0)) be the solution of (1.12) and (ut,h,Pt,h) E V1 x Qd be the solution of method (4.14) and assume the conditions of Theorems 3.22 and 3.23. Furthermore, assume that f is constant throughout the domain. Given, D M G PH,h = Pt,h + Pe,h + Pe,h + Pe.h> where satisfy (2.31) with right hand side using Pt,h satisfy (4.38), and is given in equation (4.37), there exists C such that liPPH,hllo,n CH(H + h)alult,n + CH2IPI2.n + CHhallgllo,n' liPPt,hllo,n CH(H + h)alult,n + CH2IPI2,n + CHhallgllo,n Proof: By Theorem 3.23, we must estimate aB((uuH,h,PPH.h), (fJ, il)), for which we recall, i7 = CH(TJ)IIn(CH(TJ)) and r, = To shorten notation, we define ry1 := Now, equation (4.69), which measures the lack of consistency, and the identity PK(TJt) = V' imply B((uUH,h,PPH,h), (fJ, ij)) = B((u,p), (i], ij))B((uH,h,PH,h), (fJ, ij)) 115
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L [(oun,h, V'ph(TJdPK(TJd)KV'ry)K] KETH = L (oV'ph(Ut,h), V'ph(TJt)V' L V'il)K (4.84) KETH We estimate each of these terms. First, notice the Galerkin orthogonality indicates (4.85) Next, by the proof of Theorem 3.20 (see (3.43)) and the bound on LKETH in Lemma 3.21, the relevant interpolation results, and the fact that ITJI2.n Ph,H llo,n we have the bound, Also, using the bound on from Lemma 4.7 together with the first result of Lemma 4.14, the fact that 171 := Cn( V'TJ) together with the stability of Clement 116
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interpolation, and the properties of 'fJ as the solution of (3.33) as in Lemma 3.23, L oV?ph(ryt)oV? KETH 1 ( L laph(ryt) 2 KETH :::; :::; ::S CH [ Hluh.n + )alll(uUt,h,PPt,h)IIIH] liPPH,hllo,n (4.87) Taking everything together, we find oB((uUH,h,pPH,h), (f], ij)) liPPH,hllo,n (4.88) (4.89) Applying this in the first result of Theorem 3.23, then applying the result of Theorem 4.15, there exists C > 0 such that liPPH,hllo,n ::S CH(H + h)aluh.n + CH2IPIHl + CHhall9llo,n (4.90) The second result of the theorem follows similarly when using the second result of Theorem 3.23 and part 3 of Lemma 4.7. Finally, note that the mass local mass conservation property L V7 ( Ut,h + = L g 117
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pointed out in Lemmas 3.12 and 3.13 holds for the twolevel methods in this section. This follows because even though we have computed the terms u:1 and approx imately, they still satisfy JK \7 (u:1 + = 0. Having noted this, we close this section with a result on convergence in the divergence norm. Theorem 4.17 Let (u,p) E {[H2{0)]2 n H0(div, 0)) x {H2(0) n be the solution of (1.12) and (u,,h,Pl,h) E V, x Qd be the solution of method (4.14). Given (see equation (2.37) to compute it), it follows that there exists C > 0 such that IIV (uu,,hCHiul2,n, lluu,,h:S C [(H + h)lu!I,n + Hlul2,n + + hiiYIIo,n] Proof: Following the steps of the proof of Theorem 3.26 shows the result. 4.2.5.2 Convergence results for the fully discrete full method. We now establish an estimate for the error committed by the solution to the fully discrete full method ( 4.17) in the energy norm. As was the case for the fully discrete symmetric method (4.14), this result may ultimately be used to establish convergence results in natural norms (see Theorem 4.19 and Lemma 4.21 ). Theorem 4.18 Let (u,p) E {[H1{0)]2 n H0(div, 0)) x {H2(0) n be the solution of ( 1.12), ( u1 ,h, p1,h) E V 1 x Qd be the solution of method (4.17), and (u1,h,Pl,h) be the solution of method (4.14). Under the assumption of Lemma 4.9, there exists C such that lll(uu,,h,pPl,h)IIIH:::; Ch.ja (luh,n + IIYIIo,n) +CJ(l + a)lll(uu,,h,PPl,h)lllw 118
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Proof: Notice that uUt,h = uUt,h + Ut,hUt,h := eu + Wt, and Pih,h = PPt,h + Pt,hPt,h := eP + Tt. With this, we may write Theorem 4.13 gives us a bound on the first term on the lefthand side. We must estimate the second term. For that, we use the defintion of Bh ( ) ( 4.18), the consis tency error from Lemma 4.11, and the stability of Vi( ( ) with respect to the boundary 119
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condition. = Bh((ul,hu,pl,hp), (wh rl)) +Bh((uul,h,PPt,h), (wt, rt)) +Bh((uUt,h,PPI,h), (w1, r1)) + L ap(wl) + Vrt)K :S Ch2a'Y [lultn + + L +! L KETH KETH L KETH :S Ch2a"f + + C;a L FEt:H + ( + lll(wl, :S Ch2a'Y [lultn + + C;a lll(eu, (4.92) where we have assumed 'Y = 4ff. So, we see Combining this with (4.91) we see lll(uul,h),p:S Ch2a [luli,n + + ( 1 + lll(eu, (4.93) 120
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So, we have established the theorem. As a result, we have the following theorem Theorem 4.19 Let (u,p) E ([H1(n)j2 n H0(div, n)) x (H2(n) n L5(n)) be the solution of (1.12) and (ut,h,Pt,h) E Vt x Qd be the solution of method (4.17) and assume the conditions of Theorems 3.22 and 3.23 and Lemma 4.9. Furthermore, assume that f is constant throughout the domain. There exists C > 0, such that liPPH,hlio,n :S CH(H + h)alult,n + CH2IPI2.n + CHhaligll0.w liPPt,hllo,n :S CH(H + h)alult,n + CH2IPI2.n + CHhaligll0.n Proof: Using equation ( 4.66), this follows a similar path as the proof of 4.16. Remark 4.20 The last result indicates that the error with the symmetric discrete method bounds the error with the full discrete method plus order H terms. This leads to the same optimal convergence characteristic of the symmetric discrete method. We will close this section with a result on convergence in the divergence norm. Lemma 4.21 Let (u,p) E ([H2(n)j2nHo(div, n)) X (H2(n)nL5(n)) be the solution of (1.12) and (u1,h,Pt,h) E V1 x Qd be the solution of method (4.17). Given uf, and assuming the conditions of Lemma 4.9, it follows that there exists C > 0 such that ll\7 (uUt,h:S CHiul2.n, lluUt,h:S C [(H + h)lult,n + Hlub,n + H + hll9llo.n] Proof: The reader may follow the proof of Theorem 3.31. 121
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5. Numerical Experiments We are interested in the numerical validation of the RELP methods with con tinuous and discontinuous pressure interpolation. We will test the performance of the methods on both structured and unstructured families of meshes. A typical mesh from each of these families is shown in Figure 5.2. First, we will validate the theoretically predicted convergence rates of the methods. We then move on to consider a couple of variations of the quarterfive spot problem, one with constant permeability, and the other with a checkerboard permeability. The methods (2.68) and (2.65) (those not requiring a twolevel approach), are referred to as methods with analytic M K. Methods ( 4.14) and ( 4.17), are referred to as having a nonanalytic MK. When discontinuous methods are considered, the local mass conservation property will be verified. The value will be computed using the expression JKg'V (ui + max KETh IKI (5.1) In the tables, we will refer to this value by LMC. Also, in the case an elevation plot of pressure is presented for a method using discontinuous pressure interpolation, the graph is produced by an arithmetic average over all values from triangles containing a particular vertex. In the case of twolevel methods, we are presented with the option of choosing exactly how we want to approximate at the second level. Below, results are shown for several choices at the second level. The choices are: 122
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Linear interpolation with a mesh at the second level consisting of only the original triangle. Linear interpolation with a mesh at the second level consisting of four triangles formed by linking the midpoints of edges pairwise by a line segment. See Figure 5.1 for a typical example. We see the mesh is a refinement of the original triangle. Quadratic interpolation with a mesh at the second level consisting of only the original triangle. Figure 5.1: The mesh used when using a refinement at the second level with linear interpolation. We consider these options in light of the inequality (5.2) 123
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required in the proof of Lemma 4.5. We have not computed the exact value of C0 and so it is interesting to determine the sensitivity of the methods with respect to this inequality. Finally, note that the inequality is only needed to ensure that II'VPh( vi) llo,K 2: llviiio,K holds true. This is then used to prove that the solution u1 of the method converges to the true solution. This would be one of the advantages of the twolevel methods over the analytical methods, as the latter do not have a statement that convergence of u1 is guaranteed. However, just because the inequality (5.2) does not hold does not guarantee no variables will converge. In fact, the analysis of the twolevel methods could have been carried out to show the convergence of 'Vph(ui), and no such inequality would have been required. For the numerical tests using the twolevel method presented here, we have solved the local problems locally as they are needed. However, we may reduce the computational cost of the method by choosing to invert the local matrix analytically (by hand, Mathematica, Maple, etc.) and using the resulting inverse in the calculations, meaning we simply must calculate using the indicated formula. Finally, all tests are performed for problems having righthand side f = 0. 5.1 An analytical problem We first perform a convergence validation using the test case proposed in [ 18], i.e, we set 0 = (0, 1) x (0, 1), a = 1, and we give the exact pressure p = sin(27rx) sin(27ry). The velocity is computed from Darcy's law, g is obtained from the divergence of velocity, and the boundary condition is taken to be its normal component on the boundary. 124
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0.2 0.4 o.e 0.8 Figure 5.2: A typical structured (left, H mesh. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.4 0.6 0.8 :f2 ) and unstructured (right, H i4 ) The contour plots in Figure 5.4 are representative of the solutions obtained for the methods considered below. 5.1.1 The symmetric methods 5.1.1.1 Methods with an analytic MK Inspection of Tables 5.1 and 5.2 indicates the expected performance of the methods with continuous pressure interpolation according to the theory of Chapter 3. However, we note the convergence of u1 for both types of meshes. Although this is a nice property, it is not predicted by the theory. Also, notice that Figure 5.4 shows contours of the solutions on the structured mesh, indicating good performance. In the case of discontinuous pressure interpolation, we again see all theoretically predicted convergence behavior. Note again that u1 converges, and does so with or der H2 in the case of a family of structured meshes. Moreover, Table 5.3 shows u1 to 125
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converge at order H in norm II lldiv,n although this is not theoretically predicted. No tice, however, that we lose this unexpected convergence when switching to a family of unstructured meshes. Finally, notice that the expected mass conservation property is present, as shown in Table 5.4. 10' IPI(u,NI, {Pt>,l, 10 Figure 5.3: Convergence plots using continuous pressure interpolation for problem I (structured mesh). 126
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Table 5.1: Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem I (structured mesh). I norm/ H II i I I i6 I i2 I 1 64 Utllo,n 1.5566 0.40815 0.10645 2.7804e2 7.1215e3 lu7r(udlo n 2.0220 1.0080 0.50362 0.25183 0.12592 uHilo,n 1.9129 1.1944 0.63266 0.32111 0.16118 iuuddiv,n 18.182 8.1581 4.1805 2.3360 1.2506 liP8.5212e2 2.0896e2 5.0549e3 1.2326e3 3.0603e4 JjpPHiio,n 7.4571e2 2.3306e2 6.1690e3 1.5661e3 3.9314e4 IPPtkn 1.9419 0.96058 0.50362 0.25183 0.12592 0.2 0.4 0.6 0.8 II rate I 1.95 1.00 1.00 0.93 2.01 1.99 1.00 Figure 5.4: Contour plots of the absolute value of velocity u1 (left figure) and pres sure p1 (right figure) using continuous pressure interpolation for problem 1 (structured mesh). 127
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10' 10' I ;.' c.. PP,I, 10' .' 10'' 0.99G?J' . 0.99 1.96 10. 10. 10. to 101 to 1r! Figure 5.5: Convergence plots using for the symmetric methods with analytic MK and continuous pressure interpolation for problem 1 (unstructured mesh). Table 5.2: Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem 1 (unstructured mesh). I norm/ H II I 1 1 6 I I I llu u.llo,u 0.72877 0.23827 8.0800e2 3.7565e2 1r( u1) llo,n 1.0806 0.52353 0.26235 0.12991 lluuHIIo,n 1.2944 0.71615 0.36498 0.18255 U1 11.540 7.3057 4.9394 4.3137 P1llo,n 4.7705e2 1.2250e2 2.8023e3 7.0313e4 PHIIo,n 3.4544e2 8.6819e3 2.0430e3 5.0339e4 IPPdl,U 1.3328 0.65166 0.32742 0.16273 128 1 128 1.8694e2 6.5452e2 9.2180e2 4.5570 1.7887e4 1.2856e4 8.2433e2 II rate I 1.00 0.99 0.99 0.47 1.97 1.96 0.99
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10' ni 101 _, P9,1, 10 10"1 10_,2.00v 1.oo[/7 _;; 101 s 2.0001.99 . 1.ookfjtoo 10" 10_, 10. 10_, 10' 10. 10"1 11! Figure 5.6: Convergence plots using discontinuous pressure interpolation for problem I (structured mesh). Table 5.3: Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (structured mesh). I norm/ H II I I I llu utllo.n 1.4745 0.38655 9.815le2 llu?r(udlo,n 2.0052 1.0074 0.50372 1r(ut)ufllon 2.0038 1.0072 0.50370 luuHIIo,n 1.9881 1.2353 0.65243 llu utlldiv,n 16.178 7.5389 3.6846 7r(Ut)ufldiv,n 16.145 7.5309 3.6831 liP0.11901 3.1034e2 7.8258e3 liP7.9396e2 2.4856e2 6.5553e3 IPPtlt,n 2.2415 1.1629 0.58594 129 1 32 2.4637e2 0.25184 0.25183 0.33069 1.8311 1.8307 1.9603e3 1.6605e3 0.29350 1 64 6.1656e3 0.12592 0.12591 0.16591 0.91412 0.91399 4.9032e4 4.1649e4 0.14682 II rate I 2.00 1.00 1.00 0.99 1.00 1.00 2.00 1.99 1.00
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Table 5.4: Local mass conservation for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (structured mesh). H 1/4 1/8 1/16 1/32 1/64 LMC 6.22e12 4.30ell 1.94eIO 7.97eIO 3.3le9 10 1rl I IP1>,1, 10"' I' 10' 10' 1.oov t I ' 10' 10. 1.917 10. 101 1.ooVfJ1.oo 10' 10_, 10' 10'1 10' 10' Figure 5.7: Convergence plots for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh). 130
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Table 5.5: Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh). I norm/ H II k I /6 I i2 I I lluudo,n 0.65862 0.16770 4.6298e2 1.552le2 llu7r(ut)llo.n 1.0717 0.51994 0.26142 0.12934 1r(ut)ufllo,n 1.0637 0.51838 0.26122 0.12932 lluunllo,n 1.3125 0.72916 0.37307 0.18659 llu Utlldiv,n 9.6637 4.9916 2.4965 1.3032 1r(ut)uflldiv,n 9.2838 4.6380 2.2806 1.1403 liP Ptllo,n 5.1975e2 1.332le2 3.1520e3 7.885le4 liPPnllo,n 3.5663e2 8.8029e3 2.0893e3 5.1195e4 IPPth.n 1.3721 0.65489 0.32361 0.16103 1 128 6.8179e3 6.5140e2 6.5137e2 9.4204e2 0.75324 0.57579 1.9995e4 1.3083e4 8.0987e2 Table 5.6: Local mass conservation for the symmetric method with analytic MK and discontinuous pressure interpolation for problem I (unstructured mesh). H 1/8 1/16 1/32 1/64 1/128 LMC 2.62ell 1.78eIO l.l6e9 4.77e9 2.40e8 5.1.1.2 Methods with a nonanalytic MK Notice that u1 does not converge at the expected rate when linear interpolation with no refinement is used at the second level, as indicated in Table 5.7. However, we do see convergence of V' Ph ( ut). In fact, all other variables converge at the expected rates. This indicates a sensitivity to inequality (5.2). It is interesting to note that u1 131 II rate 1.14 0.99 0.99 0.99 0.87 0.99 1.97 1.96 0.99
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appears to have the expected convergence rate on an unstructured family of meshes (when allowing that the rates may not be exact on such a family of meshes, which can't be seen as a continued refinement) as indicated in Table 5.8. However, all these issues are cleared up when either quadratic interpolation or refinement using linear interpolation are used at the second level. See Tables 5.95.12. For discontinuous pressure interpolation, we again test the convergence for several options for computing at the second level. We note that the poor convergence of u1 from the method with continuous pressure interpolation when linear interpolation with no refinement is used at the second level is rectified (compare Figures 5.7 and 5.13). All rates are equal to or better than the theoretically predicted rates of convergence. Table 5.7: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (structured mesh, linear second level). norm/ H II lluu1llon llup(u1)11o,n lluuHIIo,n llu udldiv,n liPP1llo,n liPPHIIo,n IPPd1,U 1 4 1.1983 1.8029 1.3080 15.379 0.16680 0.12191 1.8974 I 8 0.31530 0.80200 0.59683 7.2118 4.6683e2 3.3732e2 0.91025 I T6 8.5400e2 0.38414 0.29355 3.8547 1.1998e2 8.6510e3 0.44591 132 I 32 3.0799e2 0.18976 0.14623 3.7184 3.0201e3 2.1766e3 0.22163 1 64 2.1018e2 9.4587e2 7.3049e2 6.6086 7.5632e4 5.4503e4 0.11064 II rate I 0.73 1.00 1.00 2.00 2.00 1.00
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Table 5.8: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (unstructured mesh, linear second level). I nornV H II iullup(u1)11o,n lluuHIIo,n llu utlldiv,n Ptllo,n liPPHIIo,n IPP1h.n 1 8 0.51567 1.0251 0.74265 10.702 7.0844e2 5.2386e2 1.1706 1 16 0.19303 0.48885 0.36943 8.7347 1.9192e2 1.4050e2 0.58206 1 32 7.8499e2 0.23991 0.18557 7.6210 4.7309e3 3.4737e3 0.28812 1 64 3.9715e2 0.11933 9.2275e2 7.8471 1.1851e3 8.7045e4 0.14330 1 128 2.0779e2 5.9936e2 4.6656e2 8.2126 3.0058e4 2.2097e4 7.2190e2 II rate I 0.95 1.00 0.99 1.97 1.97 0.99 Table 5.9: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (structured mesh, linear second level w/ refinement). I nornV H II u1llo,n llup(ut)llo,n lluuHIIon llu u1lldiv,n liP P1llo,n liPPHIIo,n IPP1h.n 1 4 1.1968 1.1550 1.3794 15.378 0.12960 7.3530e2 1.7259 1 8 0.31290 0.30496 0.75694 7.1579 3.7150e2 1.9058e2 0.84255 133 1 16 8.0641e2 7.8172e2 0.39343 3.5106 9.5780e3 48185e3 0.41632 1 32 2. 1014e2 2.0067e2 0.19871 1.8325 2.4083e3 1.2078e3 0.20691 1 64 5.4556e3 5.1390e3 9.9583e2 0.97756 6.0238e4 3.0210e4 0.10336 II rate I 1.95 1.97 1.00 0.91 2.00 2.00 1.00
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Table 5.10: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (unstructured mesh, linear second level w/ refinement). I norm/ H II lluudlo,n llup(u1)11o,n lluunllon udldiv,n liP P1llo,n liPPnllo,n IPP1II.n 10' 10' 10_, 10_, 1.99[7 I 8 0.47497 0.47211 0.88731 9.6455 5.8128e2 3.1205e2 1.1489 10. 2.oo[Z 10" 10. 10' ro I 16 0.13892 0.13545 0.49490 5.5499 1.5635e2 8.1935e3 0.56868 ,10 I 32 4.0784e2 3.8605e2 0.24824 3.4812 3.8295e3 1.9599e3 0.28453 10' 10 101 10. I 64 1.5740e2 1.4276e2 0.12472 2.8502 9.5922e4 4.9005e4 0.14188 I I I )' 101 1 128 7.8309e3 6.9462e3 6.2800e2 3.0029 2.4356e4 1.2491e4 7.1607e2 10 II rate I 1.01 1.04 0.99 1.98 1.97 0.99 Figure 5.8: Convergence plots using continuous pressure interpolation for problem 1 (structured mesh, quadratic second level). 134
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Table 5.11: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem I (structured mesh, quadratic second level). I nonlli H II llu utllo,n llup(ut)llon llullu U1lldiv,n liP Ptllo,n liPPHIIo,n IPPdt,n 10' 10' 10' 1 4 1.1945 1.1546 1.3754 15.372 0.11259 6.2645e2 1.7716 :.: I / 10"' L 1.97 10. 10' 10"' 1 8 0.31204 0.30480 0.75663 7.1449 3.2576e2 1.554le2 0.83903 10' 16 7.9640e2 7.7699e2 0.39341 3.4663 8.4359e3 3.9153e3 0.41650 10' 10 1 32 2.0219e2 1.9631e2 0.19874 1.7260 2.1262e3 9.8072e4 0.20849 v 101 0.98v ,. 10' 1.06 10' 10' 10' 10' 1 64 5.1035e3 4.9369e3 9.9618e2 0.86648 5.3253e4 2.4528e4 0.10449 t>1>,1, 10' II rate I 1.99 1.99 1.00 0.99 2.00 2.00 1.00 Figure 5.9: Convergence plots using continuous pressure interpolation for problem I (unstructured mesh, quadratic second level). 135
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Table 5.12: Table of convergence results for the symmetric method with non analytic MK and continuous pressure interpolation for problem 1 (unstructured mesh, quadratic second level). I norm/ H II I llu uiiio,n 0.47428 llup(u1)llo.n 0.47242 lluuHIIo,n 0.88439 llu uiiidiv,fl 9.6015 liP PIIIo,n 5.2039e2 liPPHIIo,n 2.6599e2 IPP1lt.n 1.1532 I 16 0.13629 0.13368 0.49353 5.3587 l.3949e2 6.9002e3 0.56779 I 32 3.8619e2 3.6938e2 0.24780 3.2058 3.3934e3 1.6092e3 0.28477 1 64 l.4283e2 1.3080e2 0.12450 2.5148 8.4986e4 4.0178e4 0.14203 I I28 7.0145e3 6.2678e3 6.2668e2 2.6723 2.16lle4 1.0268e4 7.1805e2 II rate I 1.03 1.06 0.99 1.98 1.97 0.98 Table 5.13: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 1 (structured mesh, linear second level). norm/ H II llu uiiio,n p(ui)IIo,n lluU1 lluuHIIo,n llu uiiidiv,n llu UI liPliPPHIIo,n IPPdi,fl I 4 1.1812 1.7956 l.l820 1.3020 15.291 15.284 0.16863 0.12148 1.8996 I 8 0.30628 0.80063 0.30642 0.59827 7.1248 7.1210 4.6319e2 3.3602e2 0.82809 136 I 16 7.7597e2 0.38394 7.762le2 0.29500 3.4499 3.4487 1.1839e2 8.6175e3 0.38794 I 32 1.9468e2 0.18973 1.9473e2 0.14705 1.7007 1.7002 2.9759e3 2.1682e3 0.19025 1 64 4.8712e3 9.458le2 4.8724e3 7.347le2 0.84488 0.84465 7.4500e4 5.429le4 9.4647e2 II rate I 2.00 1.00 2.00 1.00 1.01 1.01 2.00 2.00 1.01
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Table 5.14: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (unstructured mesh, lin ear second level). I nonn/ H II lluu1ilo.n p(u.)io,n u1lluuHilon llu uddiv,n U1 liPPdlo,n liPPHIIo,n IPPd1,U I 8 0.47982 1.0229 0.47901 0.74626 9.1304 8.7952 7.3065e2 5.2730e2 1.3369 1 16 0.12698 0.48470 0.12718 0.36684 4.6283 4.3590 1.9029e2 1.3849e2 0.52920 137 I 32 3.2582e2 0.23816 32636e2 0.18496 2.2692 2.1249 4.6718e3 3.4233e3 0.24453 I 64 9.5836e3 0.11837 9.5839e3 9.1852e2 1.1510 1.0571 l.l686e3 8.5680e4 0.11971 1 128 4.0459e3 5.9392e2 4.0378e3 4.6351e2 0.62890 0.53315 2.9608e4 2.1729e4 6.0033e2 II rate I l.l8 1.00 l.l9 0.99 0.92 0.99 1.97 1.97 1.00
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Table 5.15: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (structured mesh, linear second level w/ refinement). I nonlli H II llu u1llo,n llup(ut)llo,n lluU1 ufllo,n llu UH llo,n lluu1lldiv,n U1 liP P1llo,n liPPHIIo,n IPPtlt,n 1 4 1.1829 1.1419 1.1781 1.3700 15.306 15.280 0.13286 7.2679e2 1.7077 I 8 0.30771 0.30067 0.30615 0.75495 7.1285 7.1208 3.7101e2 1.8913e2 0.79353 138 1 16 7.8005e2 7.6387e2 7.7580e2 0.39274 3.4504 3.4486 9.5106e3 4.7875e3 0.38424 I 32 1.9573e2 1.9176e2 1.9464e2 0.19845 1.7008 1.7002 2.3924e3 1.2006e3 0.19037 I 64 4.8977e3 4.7991e3 4.8704e3 9.9488e2 0.84485 0.84465 5.990Ie4 3.0038e4 9.4960e2 II rate I 2.00 2.00 2.00 1.00 1.01 1.01 2.00 2.00 1.00
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Table 5.16: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem I (unstructured mesh, lin ear second level w/ refinement). I nonlli H II lluudlo,n llup(ut)llo,n lluU1 ufllo,n uHIIo,n llu u1lldiv,n lluU1 uflldiv n P1llo,n liPPHIIo,n IPPd1,U 1 8 0.47535 0.47011 0.47079 0.88940 9.1499 8.7870 6.1376e2 3.2675e2 1.3181 1 16 0.12497 0.12409 0.12410 0.49240 4.6259 4.3559 1.5574e2 8.0835e3 0.52217 1 32 3.0774e2 3.0520e2 3.0576e2 0.24718 2.2638 2.1228 3.7789e3 1.9247e3 0.24372 1 64 7.9998e3 7.8739e3 7.9518e3 0.12416 1.1389 1.0555 9.4449e4 4.8032e4 0.11961 1 128 2.3716e3 2.2696e3 2.3568e3 6.2520e2 0.58448 0.53147 2.3929e4 1.2223e4 5.9796e2 10' I 1ri ....... ,, 10' H,,. 10' .,, IHI,I, Figure 5.10: Convergence plots using discontinuous pressure interpolation for prob lem 1 (structured mesh, quadratic second level). 139 II rate 1.75 1.79 1.75 0.99 0.96 0.99 1.98 1.97 1.00
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Table 5.17: Table of convergence results for the symmetric method with non analytic MK and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). I nornV H II llu u1llo,n llup(u1)11o,n lluu1 ufllo,n lluuHIIo,n llu uddiv,n llu U1 uflldiv,n liPliPPHIIo,n IPPtl1.n 10' 10l 10:i' 1.88 I I I I 1.98v1 .i;1.97 to to 1 4 1.1848 1.1448 1.1755 1.3314 15.320 15.277 0.11502 1 8 0.30883 0.30189 0.30600 0.73773 7.1317 7.1206 3.2482e2 6.1040e2 l.5408e2 1.6907 0.79555 10' ,. IP1>,11, IHJ. 10' ; I I I .I I 101 102 10"1 to' I 16 7.8319e2 7.6718e2 7.7560e2 0.38425 3.4510 3.4486 8.3633e3 3.9000e3 0.39165 0.99[7 1.oov?: 1.ss[7' to I 32 1.9653e2 l.9261e2 l.9460e2 0.19421 1.7008 1.7002 2.1062e3 9.7815e4 0.19505 ., to 1 64 4.9180e3 4.8203e3 4.8694e3 9.7372e2 0.84485 0.84464 5.2751e4 2.4474e4 9.7425e2 iP11,1, 10' Figure 5.11: Convergence plots using discontinuous pressure interpolation for prob lem I (structured mesh, quadratic second level). 140 II rate 2.00 2.00 2.00 1.00 1.01 1.01 2.00 2.00 1.00
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Table 5.18: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem l (unstructured mesh, quadratic second level). I nomV H II lluu1llo,n llup(u1)11o,n lluU1 ufllo,n lluuHIIo,n llu u1lldiv,n lluU1uflldiv,n liPP1llo,n liPPHIIo,n IPP1l1.n 5.1.2 The full methods 1 8 0.47584 0.47161 0.46858 0.86415 9.1669 8.7848 5.5846e2 2.8814e2 1.3371 1 16 0.12537 0.12463 0.12372 0.47907 4.6309 4.3556 l.3876e2 6.8686e3 0.53159 1 32 3.0739e2 30552e2 3.0346e2 0.24054 2.2682 2.1226 3.3272e3 l.5909e3 0.24950 1 64 7.8780e3 7.7897e3 7.7853e3 0.12081 1.1449 1.0553 8.3054e4 3.9580e4 0.12247 1 128 2.1337e3 2.0783e3 2.1098e3 6.0840e2 0.59284 0.53122 2.l045e4 1.0096e4 6.122le2 In the case that continuous pressure interpolation is used, the term dropped from the full methods in order to create the symmetric methods disappears (since [p1 ] = 0). As such, we need only consider the case of discontinuous pressure interpolation when investigating the numerical behavior of the full methods. We observe all expected rates of convergence (Tables 5.19, 5.22, 5.25, and 5.28) and mass conservation properties (e.g., Tables 5.21 and 5.24). Inspecting Table 5.20, we see a sensitivity in the method with analytic MK when o: = 1. Furthermore, when we consider Table 5.29, we see a sensitivity in the method with nonanalytic MK when o: = 10. Taken together, it seems that a safe choice for o:isO.l. 141 II rate 1.88 1.91 1.88 0.99 0.95 0.99 1.98 1.97 1.00
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Finally, convergence curves are similar to their respective curves for the symmetric methods, and so we indicate only the raw numbers and rates. 5.1.2.1 Methods with an analytic MK Table 5.19: Table of convergence results for the full method (a= 0.1) with an ana lytic MK and discontinuous pressure interpolation for problem 1 (structured mesh). I norm/ H II 1/4 I I f6 I 3 1 2 I llu udlo,n 1.4798 0.38859 9.8707e2 2.4779e2 llu1r(u1H 2.007 1.0077 0.50377 0.25184 llu1r(ur)2.0038 1.0072 0.50370 0.25183 lluuHIIon 1.9869 1.2350 0.65238 0.33068 llu u1lldiv,n 16.205 7.5442 3.6853 1.8312 lluU1 n 16.145 7.5309 3.6831 1.8307 liP P1llo,n 0.11928 3.1017e2 7.8238e3 1.9602e3 liPPHIIo,n 7.9242e2 2.4841e2 6.5543e3 1.6605e3 IPPri1,U 2.2904 1.1700 0.58687 0.29362 1 64 6.2013e3 0.12592 0.12591 0.16591 0.91412 0.91399 4.9031e4 4.1649e4 0.14683 Table 5.20: Performance of the full method with analytic MK and discontinuous pressure interpolation for problem I with different values of a (structured mesh). 1 a 11 10 1 1 1 0.1 1 0.01 1 0.001 llu1r(u1)ll 0.25174 47.030 0.25184 0.25184 0.25183 llu1r(u1)n 0.25174 0.25183 0.25183 0.25177 0.25177 liP P1llo,n l.9605e3 0.30000 1.9602e3 l.9602e3 19602e3 IPP1lr.n 0.29356 47.031 0.29362 0.29362 0.29362 142 II rate I 2.00 1.00 1.00 1.00 1.00 1.00 2.00 2.00 1.00
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Table 5.21: Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). H 1/4 1/8 1/16 1/32 1/64 LMC 6.2le12 4.3lell 1.94eJO 7.97eJO 3.31e9 Table 5.22: Table of convergence results for the full method (a = 0.1) with an analytic MK and discontinuous pressure interpolation for problem 1 (unstructured mesh). nonn/ H II lluutllo n llu1r(u1)ll llu1r(u1)ufll lluuHIIon llu Utlldiv,n llu U1 liP Ptllo,n liPPHIIo,n IPPdt,n 1 8 0.70171 1.0776 1.0688 1.3234 9.5820 9.2838 5.7889e2 4.3862e2 1.2981 1 i6 0.17661 0.52103 0.51926 0.73086 4.9273 4.6380 1.40lle2 9.8616e3 0.64071 143 l 32 4.6602e2 0.26161 0.26162 0.37331 2.4510 2.2806 3.1917e3 2.1764e3 0.31933 1 64 1.4767e2 0.12940 0.12935 0.18662 1.2599 1.1403 7.8650e4 5.1995e4 0.15857 l 128 6.0474e3 6.5165e2 6.5150e2 9.4212e2 0.69097 0.57579 1.9883e4 1.3187e4 7.976le2 II rate I 1.29 0.99 0.99 0.99 0.87 0.99 1.98 1.98 0.99
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Table 5.23: Performance of the full method with analytic MK and discontinuous pressure interpolation for problem I with different values of a (unstructured mesh). 1 a 11 10 1 1 1 0.1 1 0.01 1 0.001 llu7r(ut)ll 0.13081 0.13006 0.12940 0.12934 0.12934 llu1r(ut)0.13044 0.12910 0.12935 0.12932 0.12932 liP Ptllo,n 7.8370e4 7.8221e4 7.8650e4 7.8849e4 7.8851e4 IPPdt,U 0.15863 0.15939 0.15857 0.16100 0.16102 Table 5.24: Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (unstructured mesh). H 1/8 1/16 1/32 1/64 1/128 LMC 2.63ell 1.79el0 l.l6e9 4.78e9 2.41e8 5.1.2.2 Methods with a nonanalytic MK 144
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Table 5.25: Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). I norm/ H II 1/4 llu utllo,n 1.1951 llup(u1Ho.n 1.1559 lluu1n 1.1734 lluuHIIo,n 1.3346 llu U1 15.377 U115.275 liP P1llo,n 0.12290 liPPHIIo,n 6.2125e2 IPP1b,n 1.9457 1 8 0.31193 0.30507 0.30596 0.73834 7.1412 7.1206 3.3540e2 1.5890e2 0.83894 1 16 7.9112e2 7.7533e2 7.7560e2 0.38433 3.4525 3.4486 8.5622e3 4.0356e3 0.39758 1 32 1.9853e2 1.9466e2 1.946Ie2 0.19422 1.7011 1.7002 2.1516e3 1.0130e3 0.19580 1 64 4.9679e3 4.8716e3 4.8696e3 9.7374e2 0.84490 0.84464 5.3858e4 2.5352e4 9.7520e2 Table 5.26: Performance of the full method with nonanalytic MK and discontinuous pressure interpolation for problem I with different values of a (structured mesh). 1 a 11 10 1 1 1 0.1 1 0.01 1 0.001 llup(ut)llo,n 2.6121e2 2.5874e2 1.9466e2 1.9083e2 1.9045e2 U11.7177 1.7010 1.7002 1.7002 1.7002 liP2.8188e3 2.4131e3 2.1516e3 2.1126e3 2.1085e3 IPP1lt.n 0.19603 0.19662 0.19580 0.19576 0.19575 145 II rate I 2.00 2.00 2.00 1.00 1.01 1.01 2.00 2.00 1.01
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Table 5.27: Local mass conservation for the full method with nonanalytic MK (o: = 0.1) and discontinuous pressure interpolation for problem I (structured mesh, quadratic second level). H 1/4 1/8 1/16 1/32 1/64 LMC l.93el4 3.86el3 l.14el3 3.13e13 8.50e1l Table 5.28: Table of convergence results for the full method (o: = 0.1) with non analytic MK and discontinuous pressure interpolation for problem 1 (unstructured mesh, quadratic second level). I nonlli H II lluudlo,n llup(udllo,n lluU1ufllo n lluuHIIo,n llu u1lldiv,n lluu1uflldiv,n liP P1llo,n liPPHIIo,n IPPd1,U 1 8 0.50092 0.49072 0.48888 0.87444 9.1182 8.8051 5.8881e2 3.8454e2 1.0737 1 16 0.13068 0.12885 0.12775 0.48014 4.5982 4.3596 1.4468e2 8.0605e3 0.50359 146 1 32 3.1991e2 3.1556e2 3.1305e2 0.24070 2.2497 2.1236 3.4008e3 l.7ll2e3 0.24667 1 64 8.1865e3 8.0401e3 8.0128e3 0.12085 1.1330 1.0556 8.4085e4 4.0969e4 0.12250 1 128 2.1789e3 2.1196e3 2.1296e3 6.0853e2 0.58510 0.53124 2.1283e4 1.0362e4 6.1450e2 II rate I 1.91 1.92 1.91 0.99 0.95 0.99 1.98 1.98 1.00
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Table 5.29: Performance of the full method with nonanalytic MK and discontinuous pressure interpolation for problem I with different values of a (unstructured mesh). I a II 10 1 1 1 0.1 1 0.01 1 0.001 llu5.6011 1.42lle2 8.0401e3 1.0236e2 2.0443e2 iiu U1 uflldiv n 8.1975 1.0582 1.0556 1.0583 1.0693 liPP11io,n 4.4053e2 9.2921e4 8.4085e4 9.9379e4 3.8645e3 IPPdt,n 4.0051 0.12322 0.12250 0.12263 0.12385 Table 5.30: Local mass conservation for the full method with nonanalytic MK (a = 0.1) and discontinuous pressure interpolation for problem I (unstructured mesh, quadratic second level). H 1/8 1/16 1/32 1/64 1/128 LMC 3.19e13 3.67e12 2.44ell 1.40e8 5.95lleIO 5.2 A second analytical problem We now perform a second convergence validation. For this case, we take n = (0, 1) x (0, 1), u = 1, and we give the exact pressure p = Hx3yy3x). The velocity is computed from Darcy's law, g = 0 is obtained from the divergence of velocity, and the boundary condition is taken to be its normal component (which is no longer zero) on the boundary. We note that for the discontinuous methods, the theory predicts the method to indicate a locally massconservative velocity p( u1 ) + uf. The quantity 'V ( u1 + uf) is constant, and so in this case, 'V ( u1 + uf) = 0, (i.e.we have built a divergence free velocity field). As such, convergence of this quantity is trivial, since it is exactly what it needs to be. For this reason, the tables for the discontinuous methods show 147
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values in the divergence norm equal to the values in the 2nonn when 'V(u1 + is considered. All other rates in this section mimic the rates seen in the previous section, which are in accordance with the theoretical predictions. As such, we don't show any con vergence curves for these, choosing instead to indicate only the raw numbers and rates. The contour plots in Figure 5.12 are representative of the solutions obtained for the methods considered below. 5.2.1 The symmetric methods 5.2.1.1 Methods with an analytic MK Table 5.31: Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem 2 (structured mesh). I norm/ H II I k I 1 1 6 I f2 I llu utllo,n 1.5800e2 4.8928e3 14.137e4 3.7837e4 llu1r(u1)llo,n 6.3655e2 3.2178e2 1.6130e2 8.0682e3 lluunllon 6.1472e2 3.1038e2 1.5553e2 7.7788e3 llu u1lldiv,n 0.18078 0.14174 9.0150e2 4.9762e2 liPPtllo,n 2.6606e3 5.9207e4 1.402le4 3.475le5 liPPnllo,n 3.2226e3 8.2250e4 2.0703e4 5.1853e5 IPP1lt.n 9.0509e2 4.0030e2 1.8859e2 9.332le3 148 1 64 9.7753e5 4.0343e3 3.8895e3 2.6016e2 8.6793e6 1.2969e5 4.6595e3 II rate I 1.94 1.00 1.00 0.96 2.00 2.00 1.00
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0.9 0.9 0.8 0.8 0.7 0.7 0.6 .........__ 0.6 0.5 0.5 0.4 \ \ \ \ 0.4 0.3 \ 0.3 02 0.2 0.1 0.1 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Figure 5.12: Contour plots of the absolute value of velocity u1 (left figure) and pres sure p1 (right figure) using continuous pressure interpolation for problem 2 (structured mesh). Table 5.32: Table of convergence results for the symmetric method with analytic MK and continuous pressure interpolation for problem 2 (unstructured mesh). I norm/ H II I /6 I f2 I i4 I lluutllon 1.1026e2 5.9494e3 2.6571e3 1.4051e3 llu1r(ut)llo,n 4.2660e2 2.1209e2 1.0289e2 5.2142e3 iuuHIIo,n 4.0639e2 2.0189e2 9.7907e3 4.9618e3 Utlldiv,n 0.13831 0.16801 0.15127 0.16966 Ptllo,n 8.8979e4 2.1019e4 4.7326e5 1.2226e5 liPPHIIo,n 1.3784e3 3.3527e4 7.6917e5 1.9858e5 IPPtlt,n 4.4798e2 2.2084e2 1.0729e2 5.3920e3 149 1 64 7.1692e4 2.6174e3 2.4915e3 0.16945 3.0666e6 4.9806e6 2.7172e3 II rate I 0.98 1.00 1.00 1.99 1.99 0.99
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Table 5.33: Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 2 (structured mesh). I norm/ H II t I k I I J1 2 I lluudo,n 1.1147e2 2.7693e3 6.9121e4 1.7273e4 llu6.3621e2 3.2156e2 16123e2 8.0668e3 llu11'(ui)6.3619e2 3.2156e2 1.6123e2 8.0668e3 lluuHIIo,n 6.3616e2 3.2156e2 7.1527e4 8.0668e3 luu!ldiv 5.8553e4 1.2425e4 2.4056e5 4.4297e6 llu11'(ul)6.3619e2 3.2156e2 1.6123e2 8.0668e3 liP P1llo,n 2.8940e3 7.3840e4 1.8550e4 4.6429e5 liPPHIIo,n 2.8972e3 7.3847e4 1.8550e4 4.6429e5 IPPIII,U 6.3554e2 3.2153e2 1.6122e2 8.0668e3 150 I 64 4.3180e5 4.0341e3 4.034Ie3 4.0341e3 1.1297e6 4.034Ie3 1.16IIe5 1.1611e5 4.0341e3 II rate I 2.00 1.00 1.00 1.00 1.96 1.00 2.00 2.00 1.00
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Table 5.34: Table of convergence results for the symmetric method with analytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh). I norm/ H II I 1 16 I 312 I i4 I lluudo,n 7.2370e3 2.1630e3 9.383Je4 4.8378e4 llu7r(ut)llo,n 4.2693e2 2.1169e2 1.0258e2 5.1963e3 llu1r(ut)ufllo,n 4.2598e2 2.1156e2 1.0257e2 5.1962e3 lluuHIIon 4.2454e2 2.1145e2 1.0258e2 5.1977e3 luu1ldiv 6.4338e2 4.0715e2 2.342Je2 1.7472e2 llu1r)ut)uflldiv 4.2598e2 2.1156e2 1.0257e2 5.1962e3 liP P1llo.n 1.4702e3 3.6092e4 8.2656e5 2.1437e5 liPPHIIo,n 1.4860e3 3.5826e4 8.1631e5 2.1125e5 IPP1lt.n 4.5079e2 2.1764e2 1.049Ie2 5.3164e3 5.2.1.2 Methods with a nonanalytic MK 1 128 2.5946e4 2.6085e3 2.6085e3 2.6094e3 1.714Je2 2.6085e3 5.3996e6 5.3111e6 2.6760e3 Table 5.35: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, linear second level). II rate I 0.93 1.00 1.00 1.00 1.00 1.99 1.99 0.99 I norm/ H II 1 4 1 8 1 16 1 32 1 64 II rate I llu u1llo,n 3.0138e2 2.6841e2 2.6239e2 2.6105e2 2.6072e2 llu4.4525e2 2.2095e2 1.1028e2 5.5115e3 2.7554e3 1.00 lluuHIIo,n 4.3939e2 2.1792e2 1.0888e2 5.4689e3 2.7886e3 0.98 llu u1lldiv n 0.53594 1.1090 2.2256 4.4424 8.8675 P1llo,n 2.2781e3 5.7532e4 1.4421e4 3.6078e5 9.02IIe6 2.00 liPPH llo,n 2.1136e3 5.2994e4 1.3279e4 3.3423e5 8.5733e6 1.95 IPPd1.n 6.8916e2 3.4791e2 1.7439e2 8.7250e3 4.3633e3 1.00 151
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Table 5.36: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, linear second level). I nomV H II lluu1llo,n llup(ui)IIo,n lluuHIIo,n udldiv,n liP PIIIo,n liPPHIIo,n IPPIII,n I 8 9.3014e3 2.7636e2 2.704le2 0.18106 6.4935e4 5.3784e4 3.8276e2 1 I6 5.4436e3 1.3802e2 l3493e2 0.23764 l.6637e4 l.4078e4 1.8982e2 I 32 2.7350e3 6.7226e3 6.565le3 0.26445 3.9044e5 3.34lle5 9.2637e3 I 64 1.5916e3 3.4062e3 3.3276e3 0.31042 9.7055e6 8.3553e6 4.6526e3 _1 I28 7.8958e4 l.7102e3 l.6707e3 0.31361 2.458Ie6 2.1193e6 2.3424e3 II rate I 1.00 1.00 1.00 1.97 1.97 0.99 Table 5.37: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, linear second level w/ refinement). I nomV H II lluudlo,n llup(ui)IIo,n lluuHIIon udldiv,n liPP1llo,n liPPHIIo,n IPPII1,U I 4 1.3718e2 1.132le2 ll735e2 0.15065 2.3814e3 l.6181e3 6.8953e2 I 8 4.415le3 3.492le3 3.7685e3 0.11741 5.7786e4 4.0125e4 3.3299e2 152 I 16 1.4018e3 l.095le3 l.2739e3 8.3413e2 1.4044e4 l.0017e4 l.6llle2 I 32 4.1394e4 3.2472e4 4.443le4 5.1642e2 3.48550e5 2.5023e5 7.9831e3 I 64 l.l908e4 9.4538e5 1.7276e4 3.0342e2 8.7049e6 6.252le6 3.9857e3 II rate I 1.8 1.78 1.36 0.77 2.00 2.00 1.00
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Table 5.38: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, linear second level w/ refinement). I norm/ H II lluu1llo n llup(u1)11o,n lluuHIIo,n llu u1lldiv f! liP P1llo,n liPPHIIo,n IPP1l1,n 1 8 3.8438e3 3.4985e3 4.2096e3 8.3319e2 6.4484e4 4.2556e4 3.7673e2 1 16 2.0428e3 1.7841e3 2.1286e3 8.9796e2 1.6367e4 I.OIIle4 1.8636e2 1 32 9.9944e4 8.7631e4 1.0371e3 9.3853e2 3.8801e5 2.4261e5 9.0902e3 1 64 5.5980e4 4.8845e4 5.6628e4 0.10607 9.7156e6 6.0509e6 4.5671e3 1 128 2.7969e4 2.4562e4 2.8393e4 0.10736 2.4592e6 1.5135e6 2.3019e3 II rate I 1.00 0.99 1.00 1.98 2.00 0.99 Table 5.39: Table of convergence results for the symmetric method with nonanalytic MK and continuous pressure interpolation for problem 2 (structured mesh, quadratic second level). I norm/ H II llu utllo,n llup(u1)11o,n lluuHIIon llu u1lldiv,n liP P1llo.u liPPHIIo,n IPP1h.n 1 4 1.2957e2 1.0685e2 1.0971e2 0.12904 2.5678e3 1.8492e3 6.9482e2 1 8 3.9766e3 3.1343e3 3.3369e3 97137e2 6.2118e4 4.6292e4 3.3137e2 153 1 16 1.1775e3 9.1148e4 1.0655e3 6.4587e2 1.5198e4 1.1600e4 1.6062e2 1 32 3.2115e4 2.4721e4 3.6795e4 3.6740e2 3.7843e5 2.9028e5 7.9783e3 1 64 8.3702e5 6.4305e5 1.5042e4 1.9450e2 9.4566e6 7.2590e6 3.9853e3 II rate I 1.94 1.94 1.29 0.92 2.00 2.00 1.00
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Table 5.40: Table of convergence results for the symmetric method with non analytic MK and continuous pressure interpolation for problem 2 (unstructured mesh, quadratic second level). I norm/ H II I ...!... 16 I 32 I 64 I 128 II rate I u1llo,n 3.0720e3 1.6684e3 8.4693e4 4.7075e4 2.4222e4 0.96 llu p( ut) llo,n 2.8696e3 1.4644e3 7.4617e4 4.1260e4 2.1360e4 0.95 lluuHIIo,n 3.599le3 1.8289e3 9.2047e4 4.9804e4 2.5479e4 0.97 llu utlldiv,n 7.6344e2 7.6718e2 7.9912e2 8.9267e2 9.2552e2 liPPdo,n 7.1726e4 1.7571e4 4.1802e5 1.0479e5 2.6547e6 1.98 liPPHIIo,n 5.3492e4 1.2179e4 2.8647e5 7.1869e6 1.8028e6 2.00 IPPd1,!l 3.8209e2 1.8679e2 9.1071e3 4.5692e3 2.3057e3 0.99 Table 5.41: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, linear second level). norm/ H II 1 4 I 8 1 16 1 32 1 64 II rate I lluu1llo,n 1.1448e2 3.43lle3 1.1112e3 3.7614e4 1.3035e4 1.44 llup(ui)IIo,n 4.3499e2 2.1518e2 10717e2 5.3489e3 2.6722e3 1.00 lluU11.1404e2 3.4125e3 1.1038e3 3.7337e4 1.2935e4 1.44 lluuHIIo.n 4.3198e2 2.13552 10632e2 5.3055e3 2.6503e3 1.00 llu U1 lldiv,n 1.6089e2 7.3551e3 4.0464e3 2.5049e3 1.6539e3 0.76 llu U1 uflldiv,n 1.1404e2 3.4125e3 1.1038e3 3.7337e4 1.2935e4 1.44 liPP1llo.n 1.7657e3 4.4237e4 1.1032e4 2.7528e5 6.8741e6 2.00 liP1.7486e3 4.3513e4 1.0845e4 2.7069e5 6.7620e6 2.00 IPP1h.n 4.3784e2 2.163le2 1.0748e2 5.3568e3 2.674Ie3 1.00 154
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Table 5.42: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, lin ear second level). norm/ H lluu1llon llup(u1)11o,n lluU1lluuHIIo.n llu udldiv,n lluU1ufldiv,n liPPH!o,n IPPd1,n II 1 8 5.7163e3 2.7449e2 5.6684e3 2.7139e2 6.3162e2 5.6684e3 7.322le4 5.9885e4 3.3029e2 1 16 2.0712e3 1.3641e2 2.0572e3 1.3468e2 3.5732e2 2.0572e3 1.6284e4 1.4338e4 1.4670e2 155 1 32 6.9712e4 6.6372e3 6.9015e4 6.549le3 2.0256e2 6.9015e4 3.6264e5 3.3430e5 6.8227e3 1 64 3.2518e4 3.3512e3 3.2270e4 3.3061e3 1.2830e2 3.2270e4 8.9605e6 8.3346e6 3.4065e3 1 128 1.4728e4 1.6825e3 1.4643e4 1.6602e3 1.0766e2 1.4643e4 2.2640e6 2.1127e6 1.7105e3 II rate I 1.10 1.00 1.10 1.00 1.10 1.98 1.97 1.00
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Table 5.43: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, linear second level w/ refinement). I nomV H II llu u1llo.n llup(ul)llo,n lluu1uHIIon llu uddiv,n lluU1 liP P1llo,n liPPHIIo,n IPP1kn 1 4 9.5868e3 8.4224e3 9.5667e3 8.7004e3 1.1081e2 9.5667e3 1.8374e3 1.4654e3 4.3775e2 1 8 2.4613e3 2.165Ie3 2.4557e3 2.3580e3 3.6898e3 2.4557e3 4.6100e4 3.6490e4 2.1673e2 156 1 16 6.5070e4 5.7720e4 6.4883e4 7.1412e4 1.5680e3 6.4883e4 1.1537e4 9. 1065e5 1.0806e2 1 32 1.7976e4 1.6167e4 1.7908e4 2.5419e4 8.4565e4 1.7908e4 2.8854e5 2.2745e5 5.3978e3 1 64 5.2577e5 4.8086e5 5.2327e5 1.0560e4 5.2964e4 5.2327e5 7.2144e6 5.6834e6 2.6979e3 II rate I 1.77 1.75 1.77 1.27 0.68 1.77 2.00 2.00 1.00
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Table 5.44: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, lin ear second level w/ refinement). I norm/ H II lluudlo,n llup(ui)IIo,n llu U1 ufllo,n lluuHIIo,n llu u1lldiv n lluU1 uflldiv n liP P1ll o.n liPPHIIo,n IPP1!J.n I 8 3.5481e3 3.2154e3 3.516Ie3 3.7058e3 6.0112e2 3.516Ie3 7.4548e4 4.9507e4 3.3479e3 I 16 9.9617e4 9.0801e4 9.8836e4 1.1482e3 3.1849e2 9.8836e4 1.5114e4 9.962Ie4 1.4446e2 157 I 32 2.9596e4 2.7032e4 2.9191e4 4.0160e4 1.6943e2 2.9191e4 3.4519e5 2.3131e5 6.7798e3 1 64 1.2788e4 1.1452e4 1.2646e4 1.860Ie4 9.3028e3 1.2646e5 8.5773e6 5.6788e6 3.3902e3 I 128 5.6141e5 4.9757e5 5.5671e5 8.8403e5 6.0967e3 5.567Ie6 2.1504e6 1.4151e6 1.6990e3 II rate I 1.19 1.2 1.18 1.07 0.61 1.18 2.00 2.00 1.00
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Table 5.45: Table of convergence results for the symmetric method with non analytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, quadratic second level). J nornV H II lluudo,n llup(u1)11o,n lluu1 lluuHIIo,n lluu1lldiv,n lluU1liP P1llo,n liPPHIIo,n IPP111,U 1 4 9.4454e3 8.2698e3 9.4260e3 8.5227e3 1.0703e2 9.4260e3 2.0411e3 1.7261e3 4.752Ie2 1 8 2.3616e3 2.0580e3 2.3584e3 2.2407e3 3.0978e3 2.3584e3 5.1082e4 4.3273e4 2.3465e2 158 1 16 5.9045e4 5.1380e4 5.8978e4 6.6328e4 9.8415e4 5.8978e4 1.2772e4 1.0827e4 1.1694e2 1 32 1.4762e4 1.284Ie4 1.4747e4 2.4312e4 3.5022e4 1.4747e4 3.1931e5 2.7072e5 5.8422e3 1 64 3.6910e5 3.2102e5 3.6873e5 1.0766e4 1.3959e4 3.6873e5 7.9829e6 6.7683e6 2.9205e3 II rate J 2.00 2.00 2.00 1.18 1.33 2.00 2.00 2.00 1.00
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Table 5.46: Table of convergence results for the symmetric method with nonanalytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, quadratic second level). I nornV H II llu u1llo,n llu P( ut) llo,n lluUtn lluunllo,n llu u1lldiv,n llu U1 liP P1llo,n liPPnllo,n IPPd1,n 5.2.2 The full methods 1 8 2.9245e3 2.7054e3 29136e3 3.378le3 5.9818e2 2.9136e3 8.3005e4 6.1338e4 3.6034e2 1 16 7.5731e4 6.9735e4 7.5459e4 1.0429e3 3.1229e2 7.5459e4 1.5454e4 1.2196e4 1.5154e2 1 32 2.0497e4 1.8736e4 2.0366e4 3.9978e4 1.6533e2 2.0366e4 3.4762e5 2.7819e5 7.0866e3 1 64 7.3806e5 6.5326e5 7.3575e5 1.8860e4 9.4786e3 7.3575e5 8.6225e6 6.9254e6 3.5402e3 1 128 3.2875e5 2.8592e5 3.2877e5 9.2821e5 6.7508e3 3.2877e5 2.1605e6 1.7338e6 1.7740e3 As has been observed before in Section 5.1.2, we need only consider the case of discontinuous pressure interpolation here. We observe all expected rates of convergence and mass conservation properties. 5.2.2.1 Methods with an analytic MK 159 II rate 1.17 1.19 1.16 1.02 0.49 1.16 2.00 2.00 1.00
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Table 5.47: Table of convergence results for the full method (a = 0.1) with an ana lytic MK and discontinuous pressure interpolation for problem 2 (structured mesh). I norm/ H II 1/4 I I I f2 I llu utllo n 1.1143e2 2.769le3 6.9120e4 1.7274e4 llurr(ui)II 6.362le2 3.2156e2 l6123e2 8.0668e3 llurr(ui)ufll 6.3617e2 3.2156e2 1.6123e2 8.0668e3 lluuHIIo,n 6.3617e2 32156e2 1.6123e2 8.0668e3 llu udldiv n 5.7083e4 l.ll49e4 2.1022e5 3.8313e6 lluUtuflldiv,n 6.3617e2 3.2156e2 1.6123e2 8.0668e3 liP Ptllo,n 2.8974e3 7.3848e4 l.8550e4 4.6429e5 liPPHIIo,n 2.8974e3 7.3848e4 l.8550e4 4.6429e5 IPPtlt.n 6.3622e2 3.2156e2 1.6123e2 8.0668e3 1 64 4.3180e5 4.034le3 4.034le3 4.034le3 1.0497e6 4.034le3 l.l6lle5 I.l6lle5 4.034le3 Table 5.48: Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem 2 (structured mesh). H 1/4 1/8 1/16 1/32 1/64 LMC 1.44e13 6.33e13 3.14e12 1.32ell 8.57ell 160 II rate I 2.00 1.00 1.00 1.00 1.87 1.00 2.00 2.00 1.00
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Table 5.49: Table of convergence results for the full method (a = 0.1) with an analytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh). norm/ H llullu1r(ul)l !u1r(ul)ufll llu UH llo,n lluu1lldiv.n lluU1ufldiv,n P1lo,n liPPHIIo,n IPP1kn II 1 8 8.2244e3 4.2727e2 4.2637e2 4.2637e2 5.8814e2 4.2637e2 1.6414e3 1.6415e3 4.2739e2 1 16 2.1082e3 2.1184e2 2.1169e2 2.1169e2 3.5388e2 2.1169e2 3.7313e4 3.7314e4 2.1183e2 1 32 8.6206e4 1.0262e2 1.0259e2 1.0259e2 1.9797e2 1.0259e2 8.2882e5 8.2884e5 1.026Ie2 1 64 4.2238e4 5.1980e3 5.1971e3 5.1970e3 1.3424e2 5.1971e3 2.1268e5 2.1268e5 5.1976e3 1 128 2.2198e4 2.6095e3 2.6090e3 2.6090e3 1.2464e2 2.6090e3 5.3427e6 5.3428e6 2.6093e3 Table 5.50: Local mass conservation for the full method with an analytic MK (a = 0.1) and discontinuous pressure interpolation for problem 2 (unstructured mesh). H 1/8 1/16 1/32 1/64 1/128 LMC 6.21e13 3.21e12 1.86eII I.IOe1 0 2.6092eIO 5.2.2.2 Results with a nonanalytic M K 161 II rate I 0.93 0.99 0.99 0.99 0.11 0.99 1.99 1.99 0.99
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Table 5.51: Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem 2 (structured mesh, quadratic second level). I nonn/ H II 1/4 llu u1llo.n 9.4606e3 llup(u1)11o,n 8.2854e3 lluU19.4218e3 lluuHIIo,n 8.2927e3 llu u1lldiv,f! 1.0805e2 U19.4218e3 liP P1llo.n 2.0556e3 liPPHIIo,n 1.7399e3 IPP1lt.n 4.7754e2 1 8 2.3644e3 2.0612e3 2.3577e3 2.1005e3 3.0861e3 2.3577e3 5.1388e4 4.3636e4 2.3492e2 162 1 16 5.9101e4 5.1453e4 5.8957e4 5.5728e4 9.6317e4 5.8957e4 1.2849e4 1.0919e4 1.1697e2 1 32 1.4775e4 1.2858e4 1.4741e4 1.6802e4 3.3617e4 1.474le4 3.2123e5 2.7304e5 5.8426e3 1 64 3.6939e5 3.2144e5 3.6855e5 6.3021e5 1.3171 e4 3.6855e5 8.0310e6 6.8265e6 2.9206e3 II rate I 2.00 2.00 2.00 1.41 1.35 2.00 2.00 2.00 1.00
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Table 5.52: Table of convergence results for the full method (a = 0.1) with non analytic MK and discontinuous pressure interpolation for problem 2 (unstructured mesh, quadratic second level). I nonlli H II llu u1llo,n llup(ut)llo,n U1ufllon lluuHIIon llu u1lldiv,n lluU1liP P1llo,n liPPHIIo,n IPP1l1,n 1 8 3.7786e3 3.3895e3 3.7407e3 3.3565e3 5.6055e2 3.7407e3 9.3639e4 8.887Ie4 2.9107e2 1 16 9.6864e4 8.6966e4 9.5453e4 8.7572e4 2.9223e2 9.5453e4 1.6499e4 1.4702e4 1.4453e2 .!. 32 2.4945e4 2.2401e4 2.4525e4 2.3558e4 1.5406e2 2.4525e4 3.4586e5 2.8857e5 7.0414e3 5.3 The quarter 5spot problem, constant permeability I 64 7.7095e5 6.8309e5 7.5919e5 7.8674e5 8.6418e3 7.5919e5 8.2541e6 6.7285e6 3.5543e3 1 128 2.9889e5 2.6177e5 2.9501e5 3.336Ie5 6.0600e3 2.9501e5 2.0597e6 1.6777e6 1.786Ie3 Our final test will be with the quarter 5spot problem. This problem is carried out on the unit square, as in the previous tests. In this case, an injection well is placed at the point {0, 0) and an extraction well at {1, 1). The intent of this problem is to represent onequarter of a larger problem which consists of four injection wells placed at comers of a square domain and an extraction well at the center of the square. We will consider both the case that permeability is taken to be constant and that perme ability behaves in a checkerboard fashion (see Figure 5.37). For the tests involving the twolevel method, we will consider only quadratic solutions at the local level, as this seemed to be a top perfonner in the convergence tests considered above. 163 II rate 1.37 1.38 1.36 1.24 0.51 1.36 2.00 2.00 0.99
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Consider the case of a constant permeability Here, a = 1, and the results are tested on both the structured and unstructured meshes in Figure 5.2. 5.3.1 The symmetric methods Note the methods with discontinuous pressure interpolation have the local mass conservation property. Table 5.53: Local mass conservation for the symmetric methods using discontinuous pressure interpolation for the quarter 5spot problem with constant permeability. II Analytic MK Nonanalytic M K Mesh Structured Unstructured Structured Unstructured LMC 5.98e10 3.04e8 9.33e12 1.68e10 5.3.1.1 Methods with an analytic MK In this section, we test the performance of methods (2.68) and (2.65), considering both continuous and discontinuous pressure interpolation. The results of each test are given in four figures. In order to show the effect of enrichment, the first set of plots show contours ofthe absolute values of u1 and (Note this is simply 1r(ul) when continuous pressures are used.) The second set of plots consist of elevation and contour plots of p1 + + The contour plot is overlaid by the velocity vector field for 1r( u1 ) + and shown on top of the mesh used in calculation. In general, we see good performance on the test. On structured meshes, velocity u1 seems fairly good, but even in this case enrichment makes a marked improvement. This improvement is even more pronounced when an unstructured mesh is considered. When the full method (2.65) is used, the performance is poor for a = 1. However, the choice of 164
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o: = 0.1 alleviates this problem, indicating that, as indicated by the theory, o: should be handled carefully for the full methods (see, e.g., Theorem 2.17). The results for this case are shown in Figures 5.295.32. 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Figure 5.13: Contour plots of absolute value of velocity u1 and 1r( ui) for the method with analytic MK and continuous pressure interpolation (structured mesh). 165
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1.2 3 :l,v{ ..>.:; 0.8 0.6 0 0.4 1 0.2 2 0 ,_:_ g 0 0.2 0.4 0.5 0.2 0.6 0.8 1 1 02 0 0.2 0.4 0.6 0.8 1.2 Figure 5.14: Elevation and contour (with velocity vector field) plots of pressure p1 + for the method with analytic MK and continuous pressure interpolation (structured mesh). 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 02 0.4 0.6 0.8 Figure 5.15: Contour plots of absolute value of velocity u1 and 7!'( u1 ) for the method with analytic MK and continuous pressure interpolation (unstructured mesh). 166
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3 0.8 0.6 0 0.4 1 0.2 2 ::0.8,__:_., 0.5 0 Figure 5.16: Elevation and contour (with velocity vector field) plots of pressure p1 + for the method with analytic MK and continuous pressure interpolation (unstructured mesh). 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.4 0.6 0.8 0.4 0.6 0.8 Figure 5.17: Contour plots of absolute value of velocity u1 and 1r( u1 ) for the method with analytic M K and discontinuous pressure interpolation (structured mesh). 167
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1.2 3 2 ,;, 0.8 0.6 0 0.4 1 0.2 2 0 0 0.2 0.4 0.5 0.2 0.6 0.8 0.2 0 0.2 0.4 0.6 0.8 1.2 Figure 5.18: Elevation and contour (with velocity vector field) plots of pressure p1 + + for the method with analytic M K and discontinuous pressure interpolation (structured mesh). 0.9 0.8 0.8 0.7 0.6 0.6 0.5 0.4 0.4 0.3 0.2 0 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Figure 5.19: Contour plots of absolute value of velocity u1 and rr( u1 ) + for the method with analytic MK and discontinuous pressure interpolation (unstructured mesh). 168
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1.2 3 ,( 2 0.8 0.6 0 0.4 1 0.2 2 .. 0 iC. 3 0 0 0.8 0.2 0.2 02 0.4 0.6 0.8 1.2 Figure 5.20: Elevation and contour (with velocity vector field) plots of pressure p1 + p:1 + pf for the method with analytic MK and discontinuous pressure interpolation (unstructured mesh). 5.3.1.2 Methods with a nonanalytic MK We now consider the twolevel methods ( 4.14) and ( 4.17). As in the previous subsection, we will use both continuous and discontinuous pressure interpolation, with the same types of plots shown. The performance on the problem seems to mimic the performance of the methods considered in the previous subsection. 169
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0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.4 0.6 0.8 0.4 0.6 0.8 Figure 5.21: Contour plots of absolute value of velocity u1 and uH for the method with analytic MK and continuous pressure interpolation (structured mesh). 0.8 0.6 0.4 1 2 3 0.5 0 0.2 0.4 0.6 0.1 1 1 02 0.2 02 0.4 0.8 0.8 12 Figure 5.22: Elevation and contour (with velocity vector field) plots of pressure PH for the method with analytic MK and continuous pressure interpolation (structured mesh). 170
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0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1t\ 0 0 02 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Figure 5.23: Contour plots of absolute value of velocity u1 and uH for the method with nonanalytic MK and continuous pressure interpolation (unstructured mesh). 0.8 0.6 0.4 1 0.2 2 3 0.5 02 0.4 0.8 0.8 1.2 Figure 5.24: Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and continuous pressure interpolation (un structured mesh). 171
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0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.4 0.6 0.8 0.4 0.6 0.8 Figure 5.25: Contour plots of absolute value of velocity u1 and uH for the method with nonanalytic MK and discontinuous pressure interpolation (structured mesh). l \ .. .. ........ .. 04 1 02 2 I .,.,, 3 0.5 0 0.2 0.4 0.6 0.8 1 ""' 1 ""' 02 04 0.0 0.1 1.2 Figure 5.26: Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and discontinuous pressure interpolation (structured mesh). 172
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0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 02 0.2 0.1t\ 0.1 0 0 02 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Figure 5.27: Contour plots of absolute value of velocity u 1 and uH for the method with nonanalytic MK and discontinuous pressure interpolation (unstructured mesh). 1 2 . 0.5 0.8 0.8 o. 0.2 < .. 0.2 02 o. o.& o.a 1.2 Figure 5.28: Elevation and contour (with velocity vector field) plots of pressure PH for the method with nonanalytic MK and discontinuous pressure interpolation (un structured mesh). 5.3.2 The full methods 173
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Table 5.54: Local mass conservation for the full methods (a = 0.1) for the checker board quarter 5spot problem. II Analytic MK Nonanalytic MK Mesh Structured Unstructured Structured Unstructured LMC 5.97eIO 2.91e8 7.73e13 1.83e10 5.3.2.1 Methods with an analytic MK \ \ 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.4 0.6 0.8 0.4 0.6 0.8 Figure 5.29: Contour plots of absolute value of velocity u1 and rr( u1 ) + uf for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation (structured mesh). 174
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, 2.5 ' 1.5 0.1 0.5 0 0.5 1 1.5 t<::: 2 2.5 0 1 0.8 1 .. .. o> Figure 5.30: Elevation and contour (with velocity vector field) plots of pressure p1 + + pfD for the full method (a= 0.1) with analytic MK and discontinuous pressure interpo ation (structured mesh). I 0.9 0.9 OJ ','0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 ll3 0.4 05 0.6 0.7 0.8 0.9 1 0.4 0.6 0.8 Figure 5.31: Contour plots of absolute value of velocity u1 and rr( ui) + for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation (unstructured mesh). 175
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/ 0.8 0.8 0.4 02 0.2 02 0.4 0.6 0.8 1.2 Figure 5.32: Elevation and contour (with velocity vector field) plots of pressure p1 + + for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation (unstructured mesh). 5.3.2.2 Methods with a nonanalytic M K 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.4 0.6 0.8 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.4 0.6 0.8 Figure 5.33: Contour plots of absolute value of velocity u1 and u1 + + for the full method (a= 0.1) with nonanalytic MK (structured mesh, quadratic second level). 176
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' }2{ 0.8 0.6 0.4 1 02 2 . 0.2 0.2 02 0.4 0.6 0.8 1.2 Figure 5.34: Elevation and contour (with velocity vector field) plots of pressure p1 + + pf for the full method (a = 0.1) with nonanalytic MK (structured mesh, quadratic second level). 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Figure 5.35: Contour plots of absolute value of velocity u1 and u1 + + uf for the method with nonanalytic MK and discontinuous pressure interpolation (unstruc tured mesh, quadratic second level). 177
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1.2 4 ,.,.., 2 0.8 0.6 0 0.4 1 1 02 2 ,_, j{. 3 0 4 0.5 0.2 0 0.2 0.4 0.6 0.8 1 1 0.2 0.2 0.4 0.6 0.8 1.2 Figure 5.36: Elevation and contour (with velocity vector field) plots of pressure p1 + + pf for the method with nonanalytic MK and discontinuous pressure interpolation tunstructured mesh, quadratic second level). 5.4 The quarter 5spot problem, checkerboard permeability It is interesting to determine how well the methods perform on a problem involving a nonconstant permeability. In this case, we will consider a checkerboard permeability of the type shown in Figure 5.37. All methods considered in the previous section involving constant permeability will be tested here. We will consider only a structured mesh for these calculations. In the case that a structured mesh is used, the boundaries for regions of different permeabilities line up with edges in the mesh. Therefore, the permeability in each triangle is a welldefined constant. However, the methods involve a permeability on an edge. In the case that the edge lies on the boundary between the regions of different permeabilities, the average of the permeabilities in the two triangles incident to the edge is taken. 178
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. IV III I I II Figure 5.37: Regions in the checkerboard problem. a = 1 in zones II and IV, and a = 0.001 in zones I and III. 5.4.1 The symmetric methods We first observe from Table 5.55 that mass is conserved locally for the methods using discontinuous pressure interpolation. We observe that the performance for the methods using an analytic MK is quite good. Note that the elevation plots of the enriched pressure p1 + + show a nice performance on structured meshes, as indicated in Figures 5.38 and 5.42. Furthermore, we see that enrichment of the velocity smoothes the solution significantly as well (see Figure 5.40). We also see in Figure 5.41 that the enrichment reduces oscillations present in the polynomial solution. This makes sense, as we were not guaranteed to have control over u1 in the analysis of this method. However, the con trol of 1r( u1 ) + is guaranteed, and this seems to bear out in the numerical test. Results for the velocity for the method with a nonanalytic MK were not as good. However, we see that the effect of enrichment on the pressure solution is nearly as 179
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good as with the method with analytic MK (see Figure 5.42). Table 5.55: Local mass conservation for the symmetric methods using discontinuous pressure interpolation for the checkerboard quarter 5spot problem. Analytic MK Nonanalytic MK Mesh Structured I Unstructured Structured I Unstructured I LMC II 6.80eIO 3.02e8 4.24el0 1.78e9 5.4.1.1 Methods with an analytic MK 02 0.4 0.0 0.8 Figure 5.38: Elevation plots of pressures p1 and p1 + + for the method with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 180
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' \1 ;" I . '\ O.B 0.8 0.6 0.8 0.4 0.4 ..;.:. 0.2 \'.'. 0.2 . ',, ;>:. \:'\ , : 02 0.4 0.6 O.B 12 0.2 02 0.4 o.e o.a 12 Figure 5.39: Contour plot (with vector field of the corresponding velocity) of pres sures p1 and p1 + + for the method with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). Figure 5.40: Elevation plots of the pointwise Euclidean norrn of u 1 and u1 for the method with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 181
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1.5 0.5 ru.J Figure 5.41: Crosssections of the pointwise Euclidean nonn of u1 and u1 at x2 = .25 for the symmetric method with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 5.4.1.2 Methods with a nonanalytic MK 0 .118 0 .118 0 .116 0.06 0.04 0.04 0.02 0 .02 0 02 o. 0.6 0! 02 n 0.6 0.8 Figure 5.42: Elevation plots of pressures p1 and p1 + + for the method with nonanalytic MK (quadratic second level) and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 182
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0.8 0.8 0.6 0.6 o. 0 . 02 02 . Figure 5.43: Contour plot (with vector field of the corresponding velocity) of pres sures PI and PI + + for the method with nonanalytic MK (quadratic second level) and discontinuous pressure interpolation for the checkerboard problem (struc tured mesh). 5.4.2 The full methods We see here a repetition of the results for the symmetric methods. Table 5.56: Local mass conservation for the full methods (o: = 0.1) for the checker board quarter 5spot problem. Analytic MK Nonanalytic MK Mesh Structured I Unstructured Structured I Unstructured I LMC II 6.05eIO 2.91e8 4.22ell 9.5le10 5.4.2.1 Methods with an analytic M K 183
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0.1 0.1 0.08 0.08 0.06 0.06 0.04 0.04 0 02 0.02 0.02 0.02 0.04 0.04 0.06 0.08 0.08 0.08 0.1 0.1 02 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Figure 5.44: Elevation plots of pressures p1 and p1 + + for the method full method (a= 0.1) with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). ... 0.8 0.6 0.4 0.2 0.8 ;' ::: n\L___ '.' \} . . : : : .. : ' \\ .. . . \', : ... :_ .. : . \\ . . : 0.6 0.4 02 02 02 0.4 0.6 0.8 12 02 02 0.4 0.8 0.8 1.2 Figure 5.45: Contour plot (with vector field of the corresponding velocity) of pres sures p1 and p1 + + for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 184
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25 Figure 5.46: Elevation plots of the pointwise Euclidean norm of u1 and u1 + + uf for the full method (a = 0.1) with analytic MK for the checkerboard problem (structured mesh). 2.S 1.S o.s 0o 0.1 01 0.3 o.s 0.6 0.7 0.8 0.9 1 Figure 5.47: Crosssections ofthe pointwise Euclidean norm of u1 and u1 +u:1 +uf at x2 = .25 for the full method (a = 0.1) with analytic MK and discontinuous pressure interpolation for the checkerboard problem (structured mesh). 5.4.2.2 Methods with a nonanalytic M K 185
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O.lll 0 .(11 O .!Mi 0 .116 O .IM 0 .02 0.02 alii 0 02 0.4 0.6 0.8 Figure 5.48: Elevation plots of pressures PI and PI + + for the full method with nonanalytic MK (a = 0.1) for the checkerboard problem (structured mesh, quadratic second level). 0.8 0.8 0.6 0.6 0.4 u 02 02 Figure 5.49: Contour plot (with vector field of the corresponding velocity) of pres sures PI and PI + + for the full method with nonanalytic MK (a = 0.1) for the checkerboard problem (structured mesh, quadratic second level). 186
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6. Conclusion Although it is appealing to use the 1Ptf!P1 and !Pt/JP'fisc elements to approximate solutions to the mixed form of the Darcy problem, the pairs are unstable in the sense of the LBB condition. For this reason, one must effectively address the associated nontrivial nullspace. This has been addressed in this work in the framework of en riched methods. Indeed, the contributions ofthis work can be summarized as follows: A new enrichment strategy employing residuals, in particular edge residuals, has been introduced. This has generalized the strategy proposed for the 1Ptf!P0 element, as presented in [5] and extended in [6, 2]. The strategy has been employed to yield two new branches of oneand two level methods derived from general methods. These methods require extra care in order to consistently control the gradient of pressure, and involve jumps which are different from that present for the PGEM methods for the 1Pt/IP0 element. A numerical analysis has been performed which shows wellposedness and er ror optimality for both the oneand twolevel RELP methods, pointing out the impact of secondlevel discretization in the process. A locally conservative velocity field may be recovered with a costfree postpro cessing in the case discontinuous pressures are assumed. The theoretically predicted convergence rates and local conservation properties validated through extensive numerical tests. In addition, the methods are shown 187
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to be capable of solving the quarterfive spot problem with both homogeneous and heterogeneous permeability. A portion of these results related to method (2.50) have been presented in a corre spondence paper [23]. We also choose to highlight some nice features of the methods: they are derived via an enrichment of the standard approximation spaces; optimal convergence in the natural norms, which has also been shown for the twolevel methods; the number of degrees of freedom is not increased over those present in the IP\/1P'1 (respectively, IP't/JP1isc) approximation. the resulting linear system is smaller as compared with classical Raviart Thomas elements. the derivation indicates a mass conservative velocity A couple directions of future investigation are warranted. We have observed numerically that there seems to be a sensitivity to the ap proximation scheme used in the twolevel methods when looking at the con vergence of some variables. It would be interesting to explore this issue by considering more "enlightened" approximation schemes at the second level to see how these affect the performance of the method. In the same vein of in quiry, it would be interesting to learn more about the constant which appears to drive this sensitivity to learn how it might elucidate which schemes might be considered "enlightened." 188
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Another direction of further inquiry could be into equalorder IP'k/IP'k approx imation spaces of degree k > 1. All such elements do not satisfy the LBB condition. One possibility is to use the methods fonnulated here with such elements replacing the IP'J/IP'1 element considered here. Alternatively, meth ods could be derived by assuming these elements from the start, which would lead to methods with new tenns not present in the current fonnulation. Also, it would be interesting to investigate the approach taken here when applied to elements defined on quadrilaterals. 189
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APPENDIX A. DEFINITIONS AND THEOREMS A.l Useful Definitions Following the definition of a finite element as given in [22, 13], adapted to the particular setting for this work. Definition A.l A finite element consists of a triplet { K, P, L:} where: 1. K is a compact, connected, Lipschitz subset with nonempty interior. 2. Pis a n8hdimensional vector space of functions p : K t !Rm for some positive integerm. 3. L: is a set of n8h linear forms called local degrees of freedom acting on the elements of P and such that the linear mapping is bijective. A.2 Various Useful Inequalities It will be useful to recall the Poincare inequality [33]. Lemma A.2 SupposeD is a convex domain. Then, 'rlv E HJ(D) diamD llvllo ::; IIVvllo 7r 190
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(A similar bound with constant equal to was established in [I] for v E 1 (D) for convex domain D.) Recall the local trace inequality: given K E TH, F 8K, there exists Ct such that for all q E H1(K) (A.2) Clearly, the local trace inequality (A.2) implies the following trace inequality for vector valued functions 2 ( 1 2 2) iiv niiF:::; Ct HK llviiK + HKiviK (A.3) whenever v E [H1(K)j2. Suppose {TH} is a family of affine meshes. We will denote the transformation of local coordinates for the reference element k to element K E TH by TK. Since we only consider affine meshes, there exists matrix JK E IR.2x2 and vector bK E IR.2 such that (A.4) A mesh is said to be shaperegular when there exists a0 such that for all K E TH HK l1K =:::; ao, (A.5) TK where TK is the diameter of the largest ball that can be inscribed inK. Since for each F C 8K, TK:::; HF, the shaperegularity of the mesh implies (A.6) where Cr = ..!... ao We will also consider the following local inverse estimate, modified from the result in [22]. 191
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Lemma A.3 Let {K, P, E} be a finite element. Let l 0 be such that P C W1 00(K). Let Tn be a shaperegular family of affine meshes in IR.2 with H 1. Let 0 m l. Then, there is CJ, independent of H and K such that, for all v E PK = {poTj(1;p E P}, (A.7) A similar estimate holds when considering spaces which are related via a Piola transformation. The need for this type of transformation arises when we require the preservation of normal components. To begin, the Piola transformation from a vector space V(K) to vector space V(K) on the reference element k is defined by With this definition, we have the following inverse estimate. Lemma A.4 Let {K, P, E} be a finite element. Let P c W1 00(K). Let Tn be a shaperegular family of affine meshes in IR.2 with H 1. Then, there is CJ, indepen dent of Hand K such that,for all v E PK = { 1/Jh}(p); p E P}, (A.9) Whenever we wish to express global versions of these estimates, we will require a bound of the form: there exists Cr such that H CrHK for all H E Tn. When this hold true for a shape regular mesh, we say the mesh is quasiuniform. A.3 Some notes on an enrichment velocity. 192
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Suppose the operator PK(u1 ) is defined by problem (2.1) with vF(au1 nF) = au1 nF. This yields the problem (A.IO) The following lemma shows an orthogonality property of Mi((v) Lemma A.S The linear operator aMi( is an orthogonal projection with respect to the [L2(K)J2 inner product. More precisely,for all v E [L2(K)J2, (vw)K = 0, for all wE H0(div, K) such that '\1 w = 0 inK. Moreover, Proof: See [2]. We also have an interesting lemma regarding this PK ( ) which states that it may be used to define a finite element. In order to set this up, consider the standard linear, vectorial Lagrange finite element (K, [IP1 (K)]2 Et). The basis functions of [IP1 (K)J2 are denoted by = 1/Jkei, k E {1, 2, 3}, i E {1, 2}, where 1/Jk fonn the standard basis for the standard linear Lagrange finite element defined on triangle K and ei fonns the standard basis for Taking to be the pointwisedefined degrees of freedom for [IP1(K)j2, we may write 2 J [IP1 (K}f 3 Vt = L L (A. II) i=l k=! 193
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Here we have taken (A.I2) where [vi]i is the ith component of Vt. and xk is a vertex of the triangle K. Lemma A.6 Given K E TH. suppose PK() is defined by problem (A. 10). There ex ists a set 1:: of six linear forms acting on [p(IP\ (K))F such that { K, PK([IP\ (K)F), 1::} is a finite element. Proof: We may expand PK(vi) and PK(vt) in terms of the degrees of freedom A.l2 for vi in the following way. First, let 2 3 PK(vi) = L L i=I k=I 2 3 PK(vt) = i=I k=I It follows from problem (A. 10) Oi) must satisfy (A.I3) (A.I4) (A.I5) (A.I6) By the definition of the space PK([IP'I(K)F), it follows that if we choose w E PK([IP'I(K)J2), there will be at least some vi E [IP'I(K)J2 such that w = PK(vt). Fix such a vi and denote by 1:: a finite set consisting of linear forms which have the definition = k E {1, 2, 3}, i E {1, 2}. (A.I7) 194
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It turns out that the mapping (A.l8) where E E for k E { 1, 2, 3}, i E { 1, 2}, is bijective. We prove this by first choosing w = 0. We would like to show that v1 = 0 is the only linear vector for which PK(v1 ) = w = 0. From the problem, this assumption implies that any v1 E [IP'1{K)J2 for which PK(vJ) = 0 must satisfy the conditions '\1 VI = 0, VI nF = 0. (A.I9) It can be established that these conditions ensure v1 = 0, and we conclude that Remark A.7 This also implies that if there is Vi E VI n Ho(div, TH ), then Vi = 0. The first interpretation of this is the functions are linearly independent, and so dim(pK{[IP'1{K)j2)) = 6. In addition, we see the mapping (A.21) is injective, and therefore bijective. We further conclude the inverse mapping (A.I8) of the mapping in (A.21) is bijective. With this fact in place, we have established the result. 195
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APPENDIX B. SOME THEORY FOR OPERATOR PROBLEMS The focus of this section is to list main results for the theoretical underpinning of mixed problems. We first give some results in the classical theory, and then prove a result on a small modification to the setting of the classical theory. The author is not aware of any proof of this theorem, but does not make any claim to its uniqueness. B.l Theory for symmetric problems In order to define which conditions must be satisfied in problem ( 1.11 ), we will appeal to a more abstract fonnulation. Assume V and Q are Hilbert spaces. We take bilinear fonns a(,) E .C(V x V, IR) and b( ) E .C(V x Q, JR), and linear functionals f E V' and g E Q'. Consider the problem: Find ( u, p) E V x Q such that: forall(v,q) E V x Q. a(u, v) + b(v,p) = f(v), b(u, q) = g(q), (B. I) The theory on problem (B.I) has been thoroughly developed. We will first recall the fundamental theorems on the wellposedness, and finish with a discussion on why the weak fonn of the Darcy problem ( 1.12) is welldefined. The following discussion is pulled almost entirely from [22] and so no proofs are given. We will first define operators A : V t V' by (Aw, v)v',v = a(w, v) and B : V t Q' by (Bv, q)Q'.Q = b(v, q) Vw, v E V and Vq E Q. Let Ker(B) = {v E V : Bv = 0}. We have the following theorem 196
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Theorem B. I Problem (B.l) is wellposed if and only if 3a > 0, f a(w,v) m sup >a, wEKer(B) vEKer(B) llwllvllvllv (B.2) 'rfv E Ker(B), ('VuE Ker(B), a(u, v) = 0) ::::=::? (v = 0), and 3/3 > 0, f b(v,q) m sup > /3. qEQ vEV llvllvllqiiQ (B.3) Furthermore, we have the following a priori estimates: 1 1 llall llullv :S 11/llv + a(1 + Q jJ Q (B.4) IIPIIQ :S + )11/IIV' + (1 + )II9IIQ'" (B.5) Define now the dual operator BT : Q" = Q + V' of B by (BT q, v)v,v = (q, Bv)Q,Q' The abstract operators A and B then atlow us to recast problem (B.l) in an operator form: Find ( u, p) E V x Q which satisfy Au+ BTp= J, (B.6) Bu=g. As problem (B.l) is equivalent to (B.6), we should see conditions (B.2) and (B.3) of Theorem B.l are equivalent to the necessary (and sufficient) conditions for problem (B.6) to be wettposed. Define now the operator rrA : Ker(B) + [Ker(B)]' by (rr Aw, v)v,v = (Aw, v)v,v 'rfw, v E Ker(B). Then we have the fotlowing theorem: Theorem B.2 Problem (B.6) is wellposed if and only if: (i) rr A : Ker(B) + [Ker(B)]' is an isomorphism. 197
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(ii) B : V t Q is surjective. Condition (i) of Theorem B.2 is equivalent to conditions (B.2) and (ii) is equiva lent to condition (B.3). Owing to the particular fonn of the Darcy problem, we note the following lemma: Lemma B.3 If a(,) is coercive on Ker(B) (3a > 0, :S a(w, w)'v'w E V), then: (l") 3 > 0 1. f a(w,v) > a n wEKer(B) SUPvEKer(B) Jwlvlvlv a, (o) 3 0 0 f a(w,v) > ll a > lll vEKer(B) SUPwEKer(B) lwlvlvlv a, (iii) 'v'v E Ker(B), ('v'u E Ker(B), a(u, v) = 0) ===> (v = 0). Proof: The proof is a simple consequence of coercivity and follows the reasoning in [22], Lemma 2.8. Note that (iii) is a consequence of (ii). We close this discussion by showing the fonns of the weak Darcy problem ( 1.12) satisfy the conditions of Theorem B. I. Suppose V = H0(div, 0.) and Q = Clearly, ( 1.12) fits this formulation when we take a( w, v) = ( w, v )n and b( v, q) = (V v,q)u. and by the observation that f and g define linear functionals !() E [H0(div, 0.)]' and g() E respectively. We now note the following theorem regarding V 0 which may be found in, e.g., [22] and references therein. Theorem B.4 Assume d 2 and let 0. be a bounded open set in JRd starshaped with respect to a ball B. Then the operator V : [HJ (O.)jd t is surjective. Remark B.5 By starshaped domnin 0. with respect to ball B, we mean that for any X E n and any y E B c n. the segment joining X and y is contained in n. See [ 13, 22]. 198
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8.2 Theory for nonsymmetric problems In [16], the authors considered a problem of the type: Find (u,p) E V x Q such that: a(u, v) + b(v,p) = J(v), c(u, q) = g(q), (8.7) for all ( v, q) E V x Q. In the above, the forms b( ) and c( ) are considered to be different, though they act on the same spaces. Clearly then this is a generalization of the result for formulation (B. I). Defining B by (Bv,p) = b(v,p) and C by (Cu, q) = c( u, q ), this type of problem was shown to have a solution provided a(, ) satisifes the invertibility condition a(u, v) inf sup > o:, uEKer(B) vEKer(C) llullvllvllv (8.8) a(u, v) mf sup > o:, vEKer(B) uEKer(C) llullvllvllv (8.9) for some o: and the forms b( ) and c{, ) satisfy the conditions sup bl{lvl,lp) 2: .BtiiPIIQ Vp E Q, vEV V V (B.IO) c(u, q) sup 11112: .82IIPIIQ Vq E Q, uEV V V (8.11) for some .82 Defining operator 1rA : Ker{C) + Ker(B)' by (1rAu,v) = a(u,v) for all u E Ker(C), v E Ker(B), the above conditions on the bilinear forms are equivalent to asserting that 199
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I. 1r A is an isomorphism. 2. B is surjective. 3. C is surjective. We will consider a modification of problem (8.1) in which the bilinear fonns b(, ) and c(, )differ from one another by choosing the test and trial spaces to be different from one another. More specifically, we shall consider the problem: Find (u,p) E V x Q such that: a(u, v) + b(v,p) = f(v), c(u, q) = g(q), (8.12) for all ( v, q) E U x S. When we assume the inclusions U C V and S C Q, problem (2.15) clearly fits this framework in the case a(u, v) =(au, v)n for u E H0(div, 0), v E p(V1) EEl Ho(div, TH) b(v, p) = (\7 v,p)n for v E p(V1) EEl Ho(div, TH) and p E and c(u, q) = (\7 u, q)n for u E H0(div, 0) and q E Q1 EEl WJ... Define A : V t U' by (Au, v)u',U = a(u, v). Next, let B : V t Q' be defined with (Bv, q)Q',Q = b(v, q). Due to the reflexivity of Q, it follows that we may write (q, Bv)Q,Q' = (Bv, q)Q',Q, and so fJT : Q" = Q t V' defined by (BT q, v)v',v = (q, Bv)Q,Q' = b(v, q). We will require one more operator. Consider 6 = Blu Then 6 : U t S' has the property (6v, q)Q',Q = (Bv, q)Q',QVv E U, Vq E Q. Note that Ker(C) = Ker(B) n U. Finally, define cr : S" = S t U' by (6q, v)u',u = (q, 6v)Q,Q' = b(v, q). Then we can cast problem (8.12) as: Find (u, p) E V x S such that 200
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Find (u,p) E V x S such that Au+CTp=f, Bu=g. (B.I3) (B.I4) Define next 1rA Ker(B) + Ker(C)' by (1rAu,v)u',U (Au,v)u',U VuE Ker(B), Vv E Ker(C). We now establish equivalent conditions to the wellposedness of problem (B.I3)(B.I4). The list of conditions are familiar from the conditions for the operators related to problem (B.7) above. Theorem 8.6 Problem (B.I3)(B.I4) is wellposed if and only ifthefollowing con ditions hold: I. 1r A is an isomorphism. 2. B is surjective. 3. 6 is surjective. Proof: First, we establish that the conditions are necessary when problem (B.I3) (B.I4) is wellposed. Then we see immediately that B is surjective. We will establish that 6 is surjective by establishing both that cr is injective and Im(CT) is closed. Suppose that cr p = 0. Then for u E Ker(A) n Ker(B), 6T p = Au and Bu = 0. So, this gives us the problem: Au+ CTp=O, Bu=O. (B.I5) However, since this problem is wellposed, we have that (u,p) = (0,0). In partic ular, p = 0 and we conclude that (jT is injective. Next, let { Wn} c Im( cr) be a 201
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convergent sequence with limit w E (Ker(C))a (owing to the fact that Im(CT) c (Ker(C))a). Then Vn E N, 3pn such that 6T Pn = Wn. Let (u, p) E V x S satisfy the problem: Au+CTp=w, Bu=O. (B.I6) Since Sis closed, it follows that 3r E S such that limnoo Pn = r. Then we see from the continuity of C that Au+ 6T p = w = limnoo Wn = limnoo 6T Pn = 6T r. So, this gives us the problem Au+ cT(pr) = 0, Bu=O. (B.I7) This problem again has the unique solution (u,pr) = (0, 0), so we have p =rand u = 0. Therefore, we have found p E S such that 6T p = w, or rather w E Im(CT). Since we therefore have Im( CT) is closed and cr is injective, C must be surjective. Next, we will establish that 1r A is an isomorphism. This will be done by adapting the proof in [22] to the present circumstances. First, let h E (Ker( C))'. This may be extended to hE U' such that llhllu' = llhllu' Let (u,p) E V x Q satisfy Au+CTp=h, Bu=O. So, u E Ker(B) and we see Vv E Ker(C), (h,v) = (h,v) = (Au,v) + (CTp,v) = (1rAu,v), 202 (B.I8)
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Vv E Ker(C). Rather, we see there is au E Ker(B) which satisfies 1rAu = h for arbitrary h E (Ker( C))', and we conclude that 1r A is surjective. Next, assume 1r Au = 0. So, (Au, v) = 0 for arbitrary v E Ker(C). Therefore, Au E (Ker(C))a. But, we have established that Im( cr) = (Ker( C) )a, so 3 p E S such that Au = CT p. Therefore, we may define the problem Au+CTp=O, Bu=O. (B.I9) Since this has the unique solution ( u, p) = (0, 0) we have in particular that u = 0 and so 1r A is injective. Since we have that it's also surjective, we have established that 1r A is an isomorphism. Now, we show the conditions are sufficient for establishing the wellposedness of problem (B.I3)(B.I4). Let f E V' and g E Q' and consider the problem Au+CTp= J Bu=g (B.20) Since B is surjective, it follows that 3ug E V such that Bug = g. Owing to the fact that 1r A is an isomorphism, we may decompose u as u = + ug, where E Ker(B) is the unique element which satisfies 1r A = f Aug. Then we see that (Auf) E (Ker(C))a. Since Cis surjective, it follows that Im(CT) = (Ker(C))a, and as such 3 p E s such that cr p = f Au. The solution ( u, p) which we have constructed is therefore a solution to the problem. We now show this is the unique solution. Consider the problem Au+ CTp=O, Bu=O. 203 (B.21)
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So, we see u E Ker( B) and rr A 0. However, rr A is injective, so u = 0. So, cr p = 0. Owing to the fact that cr is injective, we therefore see p = 0. So, the solution is unique. Finally, with the assumption that the operators rr A, B, and C are surjective, we conclude their ranges are closed. In the case of C, this implies that the range of cr is closed as well. As a corollary to the Open Mapping Theorem, this implies these each has a bounded pseudoinverse, which in tum implies + u9 and p as calculated above must vary continuously with f and g (see, e.g., [32]). (See [30] for a nice exposition on pseudoinverse, in the context of degenerate saddlepoint problems.) One more important theorem will be used. In the framework of the Darcy prob lem, this relates to the relationship that must exist between the test and trial velocity spaces. Theorem B. 7 Assume a(, ) is coercive on Ker( B). Then rr A is injective if and only if Ker(C) = Ker(B). Proof: First, assume Ker(C) Ker(B). Since a(,) is coercive, the injectiv ity of rr A follows directly from item (i) of Lemma B.3 and Lemma A.39 in [22]. Next, suppose that Ker( C) s;;; Ker( B) and note the decomposition Ker( B) = Ker( C) EB (Ker( C) )j_. Clearly, (Ker( C) )j_ is not trivial. Define rr AT : Ker( C)" = Ker(C) t Ker(B)' by (rrATw,u)v'.v = (w,rrA)u,v' Since Ker(B) is a Hilbert space, we may identify (Ker(C))a = (Ker(C))j_. Since (Ker(C))j_ is nontrivial, it follows that 3u E Ker(B) such that (rrAu, v)u',V = (v, rrAu)u,u' = 0 Vv E Ker(C). This in tum implies that rr A is not injective and/or Im( rr A) is not closed in Ker( C). All that remains is to show that Im(rrA) is closed. However, the coercivity of a(,) 204
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on Ker(B) implies by Lemma B.3 that 3a > 0 such that (7rATv,w)v v inf sup a. vEKer(C)' wEKer(B) llvllvllwllv The implication is that 1r A is surjective and so Im( 1r AT) is closed. We conclude then that Im( 1r A) is closed and so 1r A is not injective. 205
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