Citation

## Material Information

Title:
Least-squares finite-element solution of the neutron transport equation in diffusive regimes
Creator:
Ressel, Klaus JuÌÂˆrgen
Place of Publication:
Denver, CO
Publisher:
Publication Date:
Language:
English
Physical Description:
viii, 101 leaves : illustrations ; 29 cm

## Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of Philosophy)
Degree Grantor:
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied mathematics

## Subjects

Subjects / Keywords:
Neutron transport theory ( lcsh )
Least squares ( lcsh )
Finite element method ( lcsh )
Finite element method ( fast )
Least squares ( fast )
Neutron transport theory ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

## Notes

Bibliography:
Includes bibliographical references (leaves 83-86).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, Applied Mathematics
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by Klaus JuÌÂˆrgen Ressel.

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|Auraria Library
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Resource Identifier:
32586019 ( OCLC )
ocm32586019
Classification:
LD1190.L622 1994d .R47 ( lcc )

Full Text
LEAST-SQUARES FINITE-ELEMENT SOLUTION OF THE
NEUTRON TRANSPORT EQUATION IN
DIFFUSIVE REGIMES
by
KLAUS JURGEN RESSEL
B. S. (Math), Universitat zu Koln, 1985
B. S. (Physics), Universitat zu Koln, 1986
M. S. (Math), Universitat zu Koln, 1991
A thesis submitted to the
Faculty of the Graduate School of the
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
1994

This thesis for the Doctor of Philosophy
degree by
Klaus Jurgen Ressel
has been approved for the
by
(? //h'fa**-/ /33
Date

Ressel, Klaus Jurgen (Ph.D., Applied Mathematics)
Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes
Thesis directed by Professor Thomas A. Manteuffel
ABSTRACT
A systematic solution approach for the neutron transport equation is considered that is
based on a Least-Squares variational formulation and includes theory for the existence and
uniqueness of the analytical as well as for the discrete solution, bounds for the discretization
error and guidance for the development of an efficient solver for the resulting discrete system.
In particular, the solution of the transport equation for diffusive regimes is studied.
In these regimes the transport equation is nearly singular and its solution becomes a solution
of a diffusion equation. Therefore, to guarantee an accurate discrete solution, a discretization
of the transport operator is needed that is at the same time a good approximation of a
diffusion operator in diffusive regimes. Only few discretizations are known that have this
property. Also, a Least-Squares discretization with piecewise linear elements in space fails
to be accurate in diffusive regimes, which is shown by means of an asymptotic expansion.
For this reason a scaling transformation is developed that is applied to the trans-
port operator prior to the discretization in order to increase the weight for the important
components of the solution in the Least-Squares functional. Not only for slab geometry but
also for x-y-z geometry it is proven that the resulting Least-Squares bilinear form is contin-
uous and V-elliptic with constants independent of the total cross section and the scattering
cross section. For a variety of discrete spaces this leads to bounds for the discretization
error that stay also valid in diffusive regimes. Thus, the Least-Squares approach in combi-
nation with the scaling transformation represents a general framework for the construction
of discretizations that are accurate in diffusive regimes.
For the discretization with piecewise linear elements in space a multigrid solver in
space was developed that gives V-cycle convergence rates in the order of 0.1 independent of
the size of the total cross section, so that one full multigrid cycle of this algorithm computes
a solution with an error in the order of the discretization error.
This abstract accurately represents the cgnten;
publication.

Signei
Thomas A. Manteuffel

CONTENTS
Acknowledgements ........................................................ v
List of Notation..................................................... vi
CHAPTER
1 INTRODUCTION AND PRELIMINARIES ....................................... 1
1.1 Introduction and Outline......................................... 1
1.1.1 Opening Remarks.............?.......................... 1
1.1.2 Outline ............................................... 1
1.2 Neutron Transport Equation and Diffusion Limit............... 2
1.2.1 Neutron Transport Equation . .............................. 2
1.2.2 Diffusion Limit.......................................... 6
1.3 Previous Work on Numerical Solution ....................... 7
1.4 Least-Squares Approach .......................................... 9
2 SLAB GEOMETRY....................................................... 13
2.1 Problems with Direct Least-Squares Approach.................. . 14
2.2 Scaling Transformation.......................................... 17
2.3 Error Bounds for NondiffUsive Regimes........................ 20
2.3.1 Continuity and V-ellipticity.............................. 20
2.3.2 Error Bounds.............................................. 27
2.4 Continuity and V-ellipticity with respect to a scaled norm... 30
2.5 Error Bounds for Diffusive Regimes........................... 38
. 3 X-Y-Z GEOMETRY .................................................... 46
3.1 Continuity and V-ellipticity ................................... 47
3.2 Spherical Harmonics ........................................... 51
3.3 Error Bounds .................................................. 54
4 MULTIGRID SOLVER AND NUMERICAL RESULTS............................... 60
4.1 Sjv-Flux and Pjv-i-Moment Equations ............................ 60
4.1.1 S^-Flux Equations......................................... 60
4.1.2 Moment Equations.......................................... 62
4.1.3 Least-Squares Discretization of the Flux and Moment Equations 64
4.2 Properties of the Least-Squares Discretization............... 66
4.3 Multigrid Solver ............................................. 75
4.3.1 Sjy Flux Equations........................................ 75
4.3.2 Moment Equations.......................................... 76
5 CONCLUSIONS........................................................ 81
5.1 Summary of Results.............................................. 81
5.2 Recommendations for Future Work................................. 82
BIBLIOGRAPHY............................................................... 83
APPENDIX
A FLUX STENCIL ......................................................... 87
B MOMENT STENCIL ..................................................... 98

ACKNOWLEDGEMENTS
I wish to thank, first and foremost, my advisor, Prof. Tom Manteuffel, for his
discuss problems in deep detail, along with his mathematical insight and creativity resulted
always in helpful answers and hints, which formed the foundation of this thesis. Secondly, I
would like to express my appreciations to Prof. Steve McCormick, who has been a consultant
to this work since its beginning and whose expertise in multilevel algorithms has proved
invaluable. I wish also to thank the remainder of my committee, Professors Jan Mandel,
Jim Morel and Tom Russell. Jan Mandel and Tom Russell taught me in their classes the
mathematical theory of Finite-Elements. From the many discussions with Jim Morel during
my stay of 9 months at Los Alamos National Laboratory I gained a lot of insight into
transport problems. I am also grateful to the Center of Nonlinear Studies at Los Alamos
National Laboratory for the financial support and the use of their facilities during this time.
Particular thanks goes to Debbie Wangerin, who guided me through the jungle of rules
imposed by the Graduate School. Moreover, I wish to express my gratitude to Dr. Gerhard
least, I would like to thank all members of the Center for Computational Mathematics of
the University of Colorado at Denver and my friends Marian Brezina, Dr. Max Lemke, Dr.
Jim Otto, Radek Tezaur and Dr. Petr Vanek for their friendship and support.

LIST OF NOTATION
For the most part, the following notational conventions are used in this thesis. Par-
ticular usage and exceptions should be clear from the context.
Scalars, Vectors and Sets
x,y,z standard space coordinates.
Zl,Zr left and right boundary of the slab.
9 polar angle with respect to z-axis.

V = cos(0).
Ot total cross section.
0's scattering cross section.
Oa = e := 1 / a T := e or Ey/a + e2, scaling parameter.
N number of Legendre Polynomials used in truncated expansion.
h mesh size.
r := 0 ,y, z).
dr := dxdydz, incremental spatial element.
n := (fij,, Qy,Qz) (cos(^) sin(0), sin(y>) sin(0), cos(0)), or in polar coordinates: D := (0, dÂ£l := ; incremental solid angle element.
m real numbers.
w complex numbers.
iz region in IR3.
diz boundary of 7Z.
\n\ Lebesgue measure of TZ.
diam(7Â£) diameter of TZ.
S1 unit sphere in M3.
Th partition of [z;, zr] or triangulation of TZ.
D = [zi,zr] x [-1,1] in ID = TZ x S1 in 3D, (Computational domain).

Multi-index Notation
/? :=G8i,/?2,/%).
\P\ Pi + 02 + 03-
nP _______ 9^L_
dyP2 <)z^'-i '
Operators
7
P
5
5_1
Â£
Â£
Q
Q
Hjv
n,
nk
Â£s
An
JL
Identity.
:= 2 f-i'dp (in 1 D)>
:= JS1 dQ (in 3 D).
P + t(I P), Scaling Transformation.
= P+1(I-P)._
unsealed Transport, operator in 1 Dor in 3 D.
:= SC, scaled Transport operator.
:= -SfiS = (1 e)(Pp + pP) + Sfj,I.
:= sns.
Projection onto space of functions, that have truncated expansion into
Legendre polynomials or spherical harmonics.
Projection onto space of functions that are piecewise polynomials of degree r on a
partition of the slab [zi,zr].
Projection onto space of functions, that are piecewise polynomials of degree r
on a given partition.
_d_
dfi
(1-p2)^
dn
Sturm-Liouville operator.
Laplacian in polar coordinates on S1.
Sn Flux operator.
Moment operator.
Functions
Mp)
p-(p)
Y,m (Q)
?(z.p)>3(Â£.0)
is
angular Flux.
Legendre Polynomial of degree l.
:= V^TTT P;(p) normalized Legendre Polynomial of degree l.
Associated Legendre Polynomial of degree l.
Spherical Harmonics.
Moment with respect to pi(fi) or Y(0)
source term in 1 D or 3 D.
:= Sq, scaled source term.
:= S~xv, for v Â£ V.
complex conjugate of v 6 V.
Vll

Function Spaces
IPr(D) polynomials on the domain D with degree smaller or equal to r.
IPr(Th) piecewise polynomials of degree < r on 7^.
C(D) space of continuous functions on D.
C{D) space of infinitely differentiable functions on D.
L2(D) space of square Lebesgue integrable functions on D.
V Hilbert space.
Vh discrete subspace of V.
V := S~1(V), where S-1 is the inverse of the scaling transformation.
Bilinear Forms
(u, v)Mn standard Euclidean inner product of Mn.
zr 1
:= f f uv dfj, dx, (in 1. D).
m -l
:= f f uv* dQdr, (in 3 D).
ns1
:= (Â£u,Â£v}.
:= (Â£Su,CSv}, double scaled form.
:= A/(,)
:= (E||< ll^H2)1/2- norm of IT*(Gi) x L2(G2),
where Gi := [zj, zr] and G2 := [1,1] in ID
and Gi := 1Z and G2 := S1 in 3D.
:= (E|/3|=* ll^ull2)1/2, semi-norm of Hk{G{) x L2(G2).
Norm associated with V.
Norm associated with V.
= (II^Sll2+M2)1/!-
Matrices
w = (wi,... ,wN)T.
1 = (!,..., 1)T N elements.
R = l-wT.
Q = diag(wi,.. ,Wjv).
M = diag(//i,.. ,Wv)-
{u,v)
a(u,v)
a(it, v)
Norms
IMI
IMIm
||fc,0
II-Ik
II-Ik
IIMII
Vlll

CHAPTER 1
INTRODUCTION AND PRELIMINARIES
1.1 Introduction and Outline
1.1.1 Opening Remarks
The Least-Squares approach represents a systematic solution technique that in-
cludes theory for wellposedness of the continuous and discrete problem, error bounds for
discretization error and guidance for the development of an efficient solver for the resulting
discrete system. Furthermore, the Least-Squares approach is a general methodology that can
produce a variety of algorithms, depending on the choice of the Least-Squares functional,
the discrete space and the boundary treatment.
For many partial differential equations (PDEs), a straightforward Least-Squares
approach is ill-advised, since it requires more smoothness of the solution than a Galerkin ap-
proach and results in a squared condition number of the discrete operator. However, problems
of first order or elliptic problems with lower order terms are converted by a Least-Squares
formulation into a self-adjoint problem. Moreover, by introducing physically meaningful new
variables and transforming the original problem into a system of equations of first order, the
smoothness requirements of the solution can be reduced and squaring the condition number
of the discrete problem is avoided. This first-order system Least-Squares (FOSLS) technique
has recently been successfully applied to the solution of general convection-diffusion problems
(Cai et al. [10], [11]) and the Stokes equation (Cai et al. [12]). In combination with a mixed
finite element discretization it can even be worthwhile to apply this technique to self-adjoint
second-order elliptic problems in. order to circumvent the Ladyzhenskaya-Babuska-Brezzi
consistency condition (Pehlivanov et al. [47]).
The subject of this thesis is the extension of the Least-Squares approach to the
solution of the neutron transport equation. This equation mathematically describes the mi-
gration of neutrons through a host medium and their interaction (absorption or scattering)
with the nuclei of the host medium after a collision. The fact that the neutron transport equa-
tion is a already first order integro-differential equation motivates a Least-Squares approach.
However, due to special properties of the transport equation, this is less straightforward than
it might appear. In diffusive regimes, where the probability for scattering is very high while
that for absorption is very low, the transport equation is nearly singular and its solution
is close to a solution of a diffusion equation. To guarantee an accurate discrete solution in
these regimes, a discretization of the transport operator is required that becomes a good
approximation of a diffusion operator in diffusive regimes. Up to the present, only a few
special discretizations have this property. Therefore, diffusive transport problems are hard
to solve. The principal part of this thesis is devoted to the extension of the Least-Squares
approach to transport problems in diffusive regimes.
1.1.2 Outline
This thesis is organized as follows: Chapter 1 continues with an introduction to the
neutron transport equation in Section 1.2, where also properties of this equation, especially

for diffusive regimes (diffusion limit), are discussed. Section 1.3 provides an overview of
previous work on numerical solution of neutron transport problems. Finally, in Section 1.4
the Least-Squares discretization and its associated standard Finite-Element theory are briefly
reviewed.
Chapter 2 deals with the application of the Least-Squares approach to one-dimensional
(slab geometry) neutron transport problems. In Section 2.1, we analyze why a Least-Squares
discretization directly applied to the transport equation is not accurate in diffusive regimes
when combined with simple discrete spaces that use piecewise linear basis functions in space.
To cure this problem, in Section 2.2 we intorduce a scaling transformation is introduced,
which plays a key role in this thesis and is applied to the transport operator prior to the dis-
cretization. For the scaled Least-Squares functional, V-ellipticity and continuity are proved
with respect to a simple un-scaled norm in Section 2.3. Here we also obtain simple bounds
for the discretization, although they are only valid for nondiffusive regimes. This is a result
of the V-ellipticity and continuity constants dependence on the total cross section at and the
absorption cross section cra, which are the coefficients in the transport equation that deter-
mine how diffusive a region is (see Section 1.2). For diffusive regimes, cr( is very large, which
causes the simple error bounds in the unsealed norm to blow up. With respect to a scaled
norm, the V-ellipticity and continuity of the Least-Squares bilinear form with constants in-
dependent of a variety of discrete spaces, the V-ellipticity and continuity with constants independent of
valid in diffusive regimes. The first class of discrete spaces consists of spaces with functions
that can be expanded into the first N normalized Legendre polynomials in angle and are
piecewise polynomials in space, whereas the second class of spaces is formed by functions
that are piecewise polynomials in space as well as in angle.
In Chapter 3 we generalize the theory of Chapter 2 to three-dimensional problems in
x-y-z geometry. In Section 3.1 we prove the V-ellipticity and continuity of the Least-Squares
bilinear form with constants independent of ot and on spherical harmonics, which are used as basis functions for the discretization in angle
(FV-approximation). Finally, in Section 3.3 we establish error bounds for the Least-Squares
discretization using spherical harmonics as basis functions in angle and piecewise polynomials
on a triangulation of the spatial domain as basis functions in space.
Chapter 4 presents numerical results for slab geometry. In Section 4.1 we introduce
the discrete ordinates Sn flux and the Pn moment equations, which are semi-discrete forms
of the transport equation. All numerical results in this chapter are based on a Least-Squares
discretization of these forms using piecewise linear or. quadratic basis elements in space. In
Section 4.2 we summarize some properties of the Least-Squares discretization. In Section 4.3
we describe the components of full multigrid solvers, that were developed for the Sn flux as
well as for the moment equations and implemented in C+-1- under use of a special designed
array class. We also include convergence rates for the multigrid solvers in this section.
In Chapter 5 we summarize our results and give some recommendations for future
work.
1.2 Neutron Transport Equation and Diffusion Limit
1.2.1 Neutron Transport Equation
Transport theory is the mathematical description of the migration of particles
through a host medium. Particles will move through a host medium in complicated zigzag
2

paths, due to repeated collisions and interactions with host particles. As a consequence of
these collisions and interactions, the particles are transported through the host medium,
which explains the name transport theory.
Transport processes can involve a variety of different types of particles such as neu-
trons, gas molecules, ions, electrons, photons, or even cars, moving through various back-
ground media. All of these processes can be described by a single unifying theory, since they
are all governed by the same type of equation, a Boltzmann transport equation (Duderstadt
and Martin [17]). The origin of this theory is the kinetic theory of gases, in which case the
transported particles and the host particles are equal to gas molecules. For this situation
Boltzmann formulated his famous nonlinear equation in 1877 (Broda [9]). Transport theory
currently plays a fundamental role in many areas of science and engineering. For instance,
the diffusion of light through stellar atmospheres (radiative transfer) and the penetration of
light through planetary atmospheres is fundamental for astrophysics. Moreover, radiation
therapy, shielding of satellite electronics, and modeling of semiconductors require transport
calculations. Further, the transport of vehicles along highways (traffic flow problems) and
the random walk of students during registration can be analyzed by transport theory.
The transport of neutrons through matter, which is considered in this thesis, is a
major part of transport theory, because of its importance for the design of nuclear reactors
and nuclear weapons. In the mathematical description of the transport of neutrons through
matter three main interactions must be taken into account. After a neutron collides with
a host nucleus, it can either get built into the nucleus (absorption), or it can be reflected,
so that its travel direction and its energy have changed (scattering), or it can cause fission,
that is, the nucleus breaks into two smaller parts and neutrons and energy are released.
Since the collision of a neutron with a nucleus and its result are uncertain, the mathematical
formulation of neutron transport is based on probability. Therefore, the quantity computed
in neutron transport is the expected density N(r, Q, E, t) of neutrons at position r moving in
direction Q with energy E at time t. Although the knowledge of the simple particle density
p(r,t), which specifies the density of neutrons independent of their travel direction and
their energy, would be sufficient for most applications, there is no equation that adequately
describes this quantity. For this reason the density is further subdivided into the density
N(r,Q,E,t) of neutrons with a specific travel direction and a certain energy. Defining
the phase space as the space spanned by the location vector and the velocity vector or
equivalently spanned by the location vector, the unit vector describing the travel direction
and the energy of a neutron, the density N can also be viewed as the phase space density.
An equation for N(r, D, E, t) may be obtained either in an abstract way from Liou-
villes theorem (Cercignani [15]), (Osborn and Yip [46]), or from a simple balance argument,
based on the neutron conservation within a small volume element of the phase space (Lewis
and Miller [34]). Both ways result in the following balance equation for the expected neutron
density N:
dN
= Q -vVN-atvN + s. (1.1)
at
Here, v = yjis the speed of a neutron with mass that has energy E. The probability
that a neutron will collide with a nucleus while traveling along a path of length dl is given
by crt dl, where (Tt{r, E) is the so called total cross section1. Since the neutron density
is, in most applications, much smaller than the density of the host nuclei, neutron-neutron
JThe reciprocal of the total cross section is called mean free path (mfp), since this is exactly the length
that a neutron can travel on the average before encountering a collision.
3

collisions can be neglected, so that crt is assumed to be independent of the neutron density
N. Otherwise, the balance equation (1.1) would be nonlinear.
The first term on the right-hand side of (1.1) represents loss of particles due to
streaming, the second term represents loss due to collisions, and s = s(r,Q, E,t) represents
both, implicit sources of neutrons due to inscattering and explicit sources. This explains
why (1.1) is a balance equation.
In the following only steady state problems are considered. Letting Â£2, E, t) :=
vN(r, Â£2, E,t) be the angular flux of neutrons, the steady state form of (1.1) becomes
Â£2 Vip(r, Â£2, E) + o-tip(r, Â£2, E) = s(r, Â£2, E). (1.2)
Further, fission as a possible interaction is excluded in the following, so that, in addition
to an external source, s includes a term ss(r,Q,E) that describes the neutrons that are
scattered into the direction Â£2 and energy E from some other direction Â£2' and some other
energy E1-. The scattering source term can be modeled as (Lewis and Miller [34, p.35])
CO
ss(r,Q,E) := J j crs(r,Q! ^Q, E'^ E)4>(r,Q',E,)dn,dE',
os1
where S1 denotes the unit sphere in IR? and the scattering cross section bility that a neutron will be scattered from Qf Â£2 and E' > E. In most cases as depends
only on the angle between Â£27 and Â£2, so crs = of Â£2' Â£2, the scattering is said to be isotropic; otherwise, it is called anisotropic.
In most applications, it is appropriate to discretize in energy to form what is known
as the multi-group-equations (Duderstadt and Martin [17, p. 407]) (Lewis and Miller [34,
p. 61])., After dividing the energy range 2 into subintervals [Ek,Ek + dE] and denoting by
ipk the flux of neutrons with energy in this interval, this discretization results in a set of so
called single-group-equations
Â£2 Vij>k + = J s1
for any flux il>k. In addition to an external source, qk now includes a term that represents
particles scattered into the energy interval [Ek,Ek + dE] from all other energy groups. In
the absence of fission, neutrons generally lose energy. In this case the multi-group-equations
can be solved by starting with the highest energy level and solving for the next lower energy
level (1.3) in turn. The computed loss from higher energy levels appears then as source in
lower energy levels. Therefore, solving the multi-group-equations reduces to solving multiple
single-group-equations.
The topic of this thesis is the numerical solution of the single-group, steady state,
isotropic neutron transport equation. The restriction to problems with isotropic scattering
is justified by the fact that, for anisotropic problems a solution technique called multigrid-in-
angle was developed by Morel and Manteuffel [44], which depends upon solving an isotropic
problem on the coarsest level (see Section 1.3). Omitting the superscript k and including
boundary conditions, (1.3) becomes for isotropic scattering:
f [Q V + trt/ crsP] ?/)(r,Â£2) = g(fjÂ£2) for (r, Â£2) 6 7Â£ x S1
\ Â£2) = g(r, Â£2) for r G dlZ A n(r) Â£2 < 0
2 Since there exists an energy E* such that the number of neutrons with energy E > E* can be neglected,
the energy range can be assumed to be a bounded interval of the form [0,iJ*].
4

which is an equation for the single-group angular flux ip(r,Q), to be determined for all points
L = (x, y, z) in a region H C IR3 and for all travel directions on the unit sphere Sl. The
operator P is defined by
PHz,Q) = J dQ,' , (1.5)
51
which is an Z.2-projection onto the space of all functions that are independent of direction
angle ft. The boundary conditions specify the inflow of particles into the region H, since
n(r) denotes the unit outgoing normal at r G dll.
In the slab geometry case, where it is assumed that = -|^ = 0 (then the
solution ip is in particular symmetric around the z-axis), so that tp(r, Q) = ip(z,/i) with
H := cos(0) where 9 is the angle between Q and the 2-axis, (1.4) reduces to
* d ^'dz + ip(zltn) = m{y) for (i > 0
ip(zr,fi) = 9r{y) for y. < 0
,
(1.6)
where the projection operator P is now given by
l
Pip(z,n) :=^ J ip(z,n')dy!
-l
(1.7)
The boundary conditions in (1.6) specify again the inflow of neutrons into the slab, since at
the left end zi of the slab the inflow directions are given by y, > 0, while at the right end zr
the inflow directions are given by \i < 0.
Figure 1.1: Computational domain for slab geometry
In neutron transport theory it is common to introduce the absorption cross sec-
tion crtt := Â£:=n-V + at(I-P) + (raP, (1.8)
and for slab geometry we have
C := + (1.9)
5

1.2.2 Diffusion Limit
Simple approximations to the transport operator, based either on ad hoc physical
assumptions such as Ficks law 3 (Lamarsh [27, pp. 125-137]) or on a Pi-approximation (Case
and Zweiffel [14, Section 8.3]), which assumes that the flux is has an expansion into the first
two Legendre Polynomials, result in a diffusion equation. This indicates that diffusion theory
is related to transport theory. Indeed, transport theory transitions into diffusion theory in
a certain asymptotic limit, called the diffusion limit.
When at * oo and 1, equations (1.4) and (1.6) respectively becomes singular.
By dividing (1.4) or (1.6) by at, it easily follows that the limit equation is (J P)ip = 0,
which is fulfilled for all functions that are independent of direction angle fi and p, respectively.
Moreover, when at * oo and ^ 1 in a certain way, the limit solution of (1.4) and (1.6),
respectively, will be a solution of a diffusion equation. This was discovered more than 20
years ago in the work by Larsen [28], Larsen and Keller [31] and Habetler and Matkowsky
[23]-
To be more specific, we consider first the slab geometry case and assume that the
slab has length 1, which can be established by a simple transformation (see Chapter 2).
Following the recent published summary of Larsen [30], we introduce a small parameter Â£
and scale the cross sections and the source in the following way:
q(z, fi) -> eq(z, fi), at - -, cra
Â£
sa,
(1.10)
where a is assumed to be 0(1). Then the transport equation becomes
Â£4>(z,/i) :=
O 1
fix- + ~{I P) + eotP
oz e
ip(z,fi) = eq(z,fi).
(1.11)
In combination with the special scaling (1.10), the diffusion limit is then defined by the
limit Â£ 0.
The physical meaning of scaling (1.10) is that the total cross section is large, so
the system is optically thick, whereas the absorption cross section and the external source
are small. We note that there are many scalings, which are capable to express this physical
situation. However scaling (1.10) stands out since the diffusion equation
= ??(*,/) (1.12)
is invariant under it in the sense that, if the substitutions (1.10) are inserted into (1.12),
then the resulting diffusion equation is independent of the scaling parameter e.
Since equation (1.11) is of singular perturbation form, general boundary layers
can be expected. Therefore, the solution is decomposed as i/>(z,fi) = ipi(z,fi) + i/'b(z, fi),
where x/)j denotes the interior solution some mean free paths away from the boundary, and
V>b denotes the solution near the boundaries. To determine i//, the following asymptotic
expansion ansatz (Friedrichs [19])
OO
o = V'(-*>?)
r=o
3 Ficks law states that the gradient of the neutron flux is proportioned to the neutron current J(r) :=
J Â£2) dQ.
S1
6

is inserted into (1.11). By equating the coefficients of different powers of e, it can be shown
(Larsen [30]) that ip is independent of angle, so ip(z, p) = ip(z), and that ip(z) is a solution
of the diffusion equation (1.12). To obtain the boundary layer part ips, an asymptotic
expansion of the boundary layer is performed, which is then matched with the interior
expansion. It follows (Habetler and Matkowsky [23]) that the boundary layer part ipB
decays exponentially with the distance from the boundary and that its leading term is
also independent of the direction angle. Altogether, this results in the following diffusion
expansion
V>(z, /i) = (z) + eR(z, /i), (1.13)
where the diffusion limit e > 0, the solution ip f the transport equation converges to the solution

This result can be directly extended to the three-dimensional case (Pomraning [48]).
With scaling (1.10), the three-dimensional transport equation becomes
Â£ip(r,Q)
0 V + -(/ P) + eaP
Â£
ip(r, Â£2) = eq(r, Â£2)
and its solution has the diffusion expansion
(1.14)
Vfc il) = where

equation
- V- JV offt
1.3 Previous Work on Numerical Solution
The number of actual neutron transport problems that can be solved in closed an-
alytical form is very small. Some examples are presented .in Wing [51, Chapter 6] and in
Duderstadt and Martin [17, Chapter 2]. Therefore, computational methods are required for
the solution of most neutron transport problems. They fall into two broad classes: deter-
ministic and stochastic. Stochastic methods, of which the Monte Carlo method is a chief
example, involves determination of the neutron distribution via random sampling of a large
number of neutrons in the system. As the number of particles in the simulation is increased,
the statistical accuracy of the resulting solution is improved. Consequently, these methods
are often prohibitively expensive, especially when high accuracy is needed. On the con-
trary, deterministic methods involve a discretization that transforms the neutron transport
equation into a finite system of algebraic equations, which can be solved by computers.
Since in this thesis a new deterministic approach is developed, we restrict the fol-
lowing overview, which is far from complete, to previous deterministic methods and begin
with the slab geometry case. For the discretization of. the angle dependence, most frequently
a discrete ordinates (Sn) method (Carlson and Lathrop [13]) is used. This approximation
assumes that the angular dependence of the solution can be expanded in a finite number
of Legendre Polynomials and a set of discrete equations results form a collocation at Gauss
quadrature points. For slab geometry, this discretization is equivalent to a Pjv-i approxi-
mation, which is a spectral Galerkin discretization, using the first N Legendre Polynomials
as basis functions (Lewis and Miller [34, Appendix D]).
7

Because of the property that the analytical solution of the transport equation is
converging in the diffusion limit to a solution of a diffusion equation in the interior of the slab,
the discretization of the spatial dependence is much more difficult. For accuracy reasons, the
discrete solution must have the same property. Therefore, this requires a discretization of the
transport operator that becomes a good approximation of a diffusion operator in diffusive
regimes. By applying the asymptotic expansion technique introduced in Section 1.2, to
the discrete solution, Larsen, Morel, and Miller [32] analyze the behavior of various special
discretizations in the diffusion limit. In the discrete case, the mesh size h has to be considered
as a second parameter besides the parameter e. Therefore, they define in their work the
following two different limits. If, for a fixed mesh size h, the discretization approximates
a diffusion operator in the limit s > 0, then the discretization is said to have the correct
thick diffusion limit. On the other hand, if the mesh size h varies linearly with e in the limit
Â£ 0 and this limit results in a consistent discretization of a diffusion operator, then the
discretization is said to have the correct intermediate diffusion limit.
Since standard finite difference discretizations, such as upwind differencing, fail to
have a correct thick diffusion limit, special discretizations have been developed that behave
correctly in diffusive regimes. Among them are the Diamond difference scheme and the
difference schemes of Lund-Wilson and of Castor, which, according to the analysis of Larsen,
Morel, and Miller [32], give the correct thick diffusion limit for the cell average flux.
Moreover, Finite-Element discretizations have been applied for spatial discretiza-
tion of the neutron transport equation in different ways. The direct Galerkin approach to
the first order integro-diiferential form (1.6) of the transport equation, as considered first
theoretically by Ukai [50] and numerically by Martin [40], does not have the correct behav-
ior in the diffusion limit, except when special discontinuous finite elements are used. The
use of discontinuous finite elements results, for example, in the Linear Discontinuous (LD)
discretization (Alcouffe et al. [2]) and the Modified Linear Discontinuous (MLD) scheme
(Larsen and Morel [33]). MLD has the additional property that with a suitable fine spatial
mesh, it can resolve boundary.layers at exterior boundaries or at interior boundaries between
media with different material cross sections.
Further, a variety of Ritz variational formulations have been proposed (see Kaplan
and Davis [26] for a summary). They have the self-adjoint second-order even-parity4 form
of the transport equation as its Euler equation and lead, independent of the choice of the
discrete Finite-Element space, to correct diffusion limit discretizations. However, the even-
parity form of the transport equation is only valid for nonvacuum regions and becomes very
tedious for anisotropic scattering or anisotropic sources (Lewis and Miller [34, p. 260]).
For the solution of the discrete system, a simple splitting iteration known as source
iteration or transport sweep has been used in the past. Because of the slow convergence of
this iteration for diffusive regimes (convergence factor 1 0(A,-)), the Diffusion Synthetic
Acceleration (DSA) method (Larsen [29]) was developed, which uses a diffusion approxima-
tion to accelerate the source iteration. By spectral analysis, Faber and Manteuffel [18] have
shown why this method is successful for problems with isotropic scattering. However, for
problems with anisotropic scattering, DSA is less effective.
Moreover, multigrid methods have been employed for the solution of discrete neu-
tron transport problems. For the LD scheme, a multigrid algorithm in space was developed
by Barnett, Morel and Harris [4] which proved to be effective even for highly anisotropic
4 It can be shown [35] by a certain transformation that the even-parity form is closely related to the
Least-Squares formulation considered here.
8

problems. For isotropic problems, this algorithm is competitive with DSA, although it uses
an expensive block-smoothing. The multigrid algorithm in space of ManteufFel, McCormick,
Morel, Oliveira and Yang [36] for isotropic problems, discretized in space by the MLD scheme,
employs a special operator induced interpolation and has been ported very efficiently to a
parallel architecture [37]. For anisotropic problems a technique called multigrid-in-angle was
developed by Morel and Manteuffel [44]. This scheme involves a shifted transport sweep to
attenuate the error in the upper half of the moments, so that the remaining error can be
approximated by the solution of a problem discretized in angle based on only the lower half
of the moments. Recursive application of this procedure leads to an isotropic problem on
the coarsest level, which can be solved by a multigrid method in space.
For higher dimensional problems, the discretization of the angle dependence also
becomes a problem. For problems with isolated sources in a strongly absorbing medium,
anomalies in the flux distribution, called ray effects (Lewis and Miller [34, p. 194]), are likely
to arise in combination with a discrete ordinates (Sjv) discretization. The 5jv discretization
causes a loss of rotational invariance, since this discretization transforms the fully rotational
invariant transport equation into a set of coupled equations that are at most invariant under
few discrete rotations. Thus, an azimuthally uniform flux, for example, is approximated by a
set of 6-functions at discrete angles, which can be very poor if the number of discrete angles
is not sufficiently large. One potential remedy is a Pn discretization, which is a spectral
Galerkin method using spherical harmonics as basis functions. This discretization results in
a fully rotational invariant discrete problem. However, the coupling of the discrete equations
is complicated and the treatment of boundary conditions is less straight forward.
As in the one-dimensional case, for higher dimensions the discretization in space
must have the correct behavior in the diffusion limit in order to obtain accurate discrete
solutions for diffusive regimes. The direct extension of the appropriate one-dimensional
discretizations is complicated, however. Bogers, Larsen and Adams [7] have shown that the
linear discontinuous (LD) finite element discretization on rectangles does not yield a correct
diffusion limit discretization, whereas the MLD discretization does. However, the efficient
solution of the discrete system resulting from the MLD discretization is an open problem.
Applying a similar multigrid algorithm, which was developed by Manteuffel et al. [36] for
the one-dimensional case, would require the extension of the operator induced interpolation
to higher dimensions, which is complicated. Morel et al. are in the process of developing a
method for three space dimensions based on the even-parity form of the transport equation
and using a Pjy discretization in angle.
We conclude that an arsenal of highly specialized computational methods exists,
whose design is adapted for particular transport problems. However, there is lack of a general
systematic solution approach that includes existence theory of the analytic and discrete
solution, error bounds for the error of discretization and guidance for the development of an
efficient solver of the resulting discrete problem. Especially for higher dimensional problems,
such an approach seems to be needed.
1.4 Least-Squares Approach
In this section, we introduce a systematic solution approach to the neutron trans-
port equation that relies on a Least-Squares self-adjoint variational formulation of (1.4), and
we summarize the associated standard Finite-Element theory. The Least-Squares approach
can be considered as a systematic solution approach, since it includes theory for the existence
and uniqueness of the analytical and discrete problem, as well as bounds of the discretization
error for a whole class of Finite-Element spaces. Furthermore, this approach will guide the
9

development of a Multilevel Projection Method (McCormick [43]) for the efficient solution
of the resulting discrete system.
A Least-Squares Finite-Element discretization with piecewise linear basis functions
in space directly applied to (1.4) does not have the correct behavior in the diffusion limit
(see Section 2.1). For this reason, the scaling-transformation [P + t(I P)], with parameter
t Â£ IR+ specified later, is applied to the transport operator prior to the discretization:
C := [P + T(I-P)][Q-V + atI-asP]
= P(Q.-Â¥.) + T(I~P){n-V) + T(Tt(I-P) + ((Tt-as)P (1.17)
= P(Q Y) + r(J P)(Q V) + rat(I -P) + aaP,
where in the last equation (at as) was Substituted by the absorption scattering cross section
aa. In this transformed operator, the Least-Squares variational formulation of (1.4) is given
by
minP(V), with F(ip) := f [\Â£ip(r,Q qa(r,Â£i)]2 dQdr, (1.18)
V J J
n
with qs = Sq. The Hilbert space V with underlying norm J| ||v will be specified later.
A necessary condition for ip Â£ V to be a minimizer of the functional F in (1.18)
is that the first variation (Gateaux derivative) of F vanishes at ip for all admissible v Â£ V,
resulting in the problem: find ip Â£ V such that
a(ip,v) := = //
m s1 n s1
qs Cv dFldr Vu Â£ V.
(1.19)
The essential part of the theory is to show that the symmetric bilinear form a(*, *) in (1.19)
is V-elliptic i.e., there exists a constant Ce> 0 such that, for all v Â£ V,
a(v,v)> Ce |M|v, (1-20)
and continuous-, i.e., there exists a constant Cc> 0 such that, for every u, v Â£ V,
|a(u,t>)| < Cc |||k IHk-
(1.21)
The proof of the continuity is straightforward, but the proof of the V-ellipticity is difficult
and tricky.
Denote the standard inner product and associated norm of L2(72. xS1) by
(
-//
u v* dQdr;
n s1
[[(/.[[ := y/(u, u) Vu,vÂ£ L2(TZ x S1),
where v* is the complex conjugate of v. Using (1.20), (1.21) and the assumption that
Qs(l,^Q Â£ L2(1Z x S1), which ensures that the functional

qs Cv dCldr
71 s1
is bounded (|/(-u)| < Cc^2||g3|| ||t>||v ), then the Lcix-Milgram Lemma (Ciarlet and Lions
[16, p. 29]) can be applied. It follows that problem (1.19) is well posed in the sense that its
solution exists, is unique and depends continuously on the data qs. The latter follows from
Ce\\ip\\v < a(ip,ip) = l(ip) < Cc1/2||g5|| ||V-[k,
10

so
pl/2
MW < ~^\ks\\.
For the Least-Squares Finite-Element discretization of (1.19), the Hilbert space V
is replaced by a finite-dimensional subspace Vh C V, and (1.19) becomes: find fa e Vh
such that
a{fa,vh) = l(vh) Vvh E Vh. (1-22)
The existence and uniqueness of a solution fa Â£ Vh of the discrete problem (1.22) follows
again from the Lemma of Lax-Milgram since, as a finite-dimensional space, Vh is a closed
subspace of the Hilbert space V and is, therefore, also a Hilbert space with respect to the
inner product of V restricted to Vh.
By subtracting (1.22) from (1.19), it follows immediately that the error is orthogonal
to Vh with respect to the bilinear form a(-, ):
- fa,Vh) = 0 Vuj, Â£ Vb. (1-23)
The Cauchy-Schwarz inequality and (1.23) lead directly to Ceas Lemma (Brenner
and Scott [8, p.62]):
ai^-fa^-^^^a^-Vh^-Vk) Vvh Â£Vh or
fc~
\W i>h\\v < \ -pr- minJlV-t'hlk,
V vhÂ£Vh
(1.24)
with the use of the V-ellipticity (1.20) and the continuity (1.21). By (1.24), the problem of
finding an estimate of the error is therefore reduced to estimating min ||^> ^>h||y. These
vhÂ£Vh
kinds of estimates are provided by approximation theory for a wide class of spaces Vh. For
example, when we consider for simplicity only a semi-discretization in space where Vh is
formed by piecewise polynomials of degree r, V = Hm(JZ) x L2(Sl), Vh C V, and the exact
solution is in Hr+1(7l) x L2(Sl), it can be proved (Ciarlet and Lions [16, Theorem 16.2])
that
min \\'ip-Vh\\m,o vkevh
where h is the maximum mesh size of the triangulation of 12, used and
M|m,o
Y, f J \Dav\2dtldr
1/2
\i>\
fc+i,o :=
. M< n s1
1 1/2
Y j j \DaTj>\2dQ,dr
|a|=fc+l si
denote the standard Sobolev norm and semi-norm (Adams [1]), respectively. Here, we use
the standard multi-index notation
D^v :=
d^v
dxPldyfodzP*
\P\ :=EA-
i=l
for j3 := (Pi,P2,p3).
11

For Vh formed by piecewise polynomials of degree r, the combination of (1-24) and
(1.25) results in the overall error bound
U MW V
The crucial point here is that we have shown V-ellipticity (1.20) and continuity
(1.21) with respect to a weighted norm with constants Ce and Cc independent of crt and
cr0, so that an error bound similar to (1.26) for a discretization in space and in angle holds
independent of the size of crt and cra. Hence, the Least-Squares Finite-Element discretization
of the scaled transport operator with piecewise polynomials of degree r > 1 as basis functions
yields an accurate discrete solution even in the diffusion limit.
12

CHAPTER 2
SLAB GEOMETRY
The Least-Squares approach is applied in this chapter to the one dimensional (slab
geometry) neutron transport equation (1.6). Throughout this chapter, we assume without
loss of generality the following:
1) The total scattering cross section is constant in space, so o-t(z) = crt. This can be
established by the transformation
Z
f(Tt(s)ds
*'=7T---------- (2-1)
f crt(s) ds
z\
The transport equation then becomes
+ rti1 ~ p) + with
ZT Z j* Zf
*1 = J at(s) ds, 2| Zl Zl
2) The slab has length T, so (zr z\) = 1. If the transformation (2.1) was already
applied, this is directly fulfilled; otherwise, this can be established with the simple
transformation z" = This changes o" (zr zi)
3) We impose homogeneous (vacuum) boundary, conditions, so g\{n) = 0 and gr{n) = 0
in (1.6). This can be done in the following way. Define
gi(fi) for (j, > 0
gr(fi) for n<0
Then, clearly, if>t(z,fj.) 6 Jf1([z;, zr]) x L2([1,1]), so that Ctpt is well defined and
we can solve the problem Ci})o = q C^b with homogeneous boundary conditions.
The original solution is then given by ip = ipo + Vv
In this chapter, let D := [zi, zr] x [1,1] and let
i
uvd/idz and ||u|| := \/(u, u)
zi -l
denote the standard inner product and associated norm of L2(D).

2.1 Problems with Direct Least-Squares Approach
In the following, we give an explanation as to why a Least-Squares Finite-Element
discretization applied to (1.6) using piecewise linear basis functions in space does not, in
general, yield a correct diffusion limit discretization. We recall that the Least-Squares vari-
ational formulation of (1.6) is given by
Jr 1
min
in F(>!>), with := J J fi) eq(z, fi) dfidx, (2.2)
Z\ -1
and
f d'v
V := < v(z,fi) G L2(D) : fi-^- G L2(D),v(zi,fi) = 0 for ft > 0, v(zr,ft) = 0 for fi < 0
In (2.2) we used the parameterized form (1.11) of the transport equation, since it is better
suited for a diffusion limit analysis.
For the discretization of (2.2), the minimization of the Least-Squares functional is
restricted to a finite-dimensional subspace Vh C V. Without loss of generality, in the follow-
ing analysis for the discretization in angle we use a Pi approximation, which assumes that
the angle dependence of the solution has an expansion into the first two Legendre Polyno-
mials. One reason for this is that a semi discretization only in angle by a Pi approximation
results in a diffusion equation [14, Section 8.3]. Second, the behavior of the discretization in
diffusive regimes, where according to (1.3) the exact solution is nearly independent of angle,
is analyzed here; thus, a Pi approximation allowing a linear dependence in angle is sufficient.
For the discretization in space, we use piecewise polynomials on a partition Th of the slab.
Altogether, this results in the discrete space
Vh := {vh G C(D) : vh(z,fi) =.0(z) + fii(a;), where 0,i G TPr{Th)\
Vh(zi,fi) = 0 for fi > 0, vh(zr,(j,) = 0 for /i < 0 } , ' '
where lPr(Th) denotes the space of piecewise polynomials.of degree < r on the partition Th
of the slab.
By the asymptotic expansion introduced in Section 1.2, the minimizer of the Least-
Squares functional can be characterized as follows.
Theorem 2.1 (Characterization of Least-Squares minimizer)
Let the Least-Squares functional F and the discrete space Vh be given as defined in (2.2)
and (2.3) respectively. Suppose tj>h G Vh minimizes F restricted to Vh. Suppose further
that e < 1 and that iph has. the asymptotic expansion in e given by
i>h{z,y) = eQ{z) +n\{z), with
= #(z) :=
v-0 z/=0
(2.4)
where rjv,5v G lPr(Th) are independent of parameter e for all v. We then have:
(i) 50{z) = 0.
(ii) %(z) = -h{z).
14

(iii) Let
Uh := {Vo Â£ IPr{Th) : Vo(zl) = ^Io(zr) = 0, rjo fulfills (ii) for some i$i Â£ IPr(Th)} . Then for all tjq Â£Uh: J ^vWo + Wo dz = *T J qr)o dz. Z\ Zl Proof. We first prove (i). Using expansion (2.4) in (1.11) we have Zij>h = j [Mo] + m'o + ft + Mi] + 0(e), and, therefore, with F(1>k) = Â£ evFv(iph), v--2 Zr F-2(i>h) = | J 6o(z) dz, Zl Zr F-ityh) = ^ J%(z)6o(z) + S0(z)61(z)dz, (2.5) and Fv(%l>h) independent of e for v > 0. For e < 1, it is possible to bound Jr('0/!) from above independent of Â£ by F(i>h) since tph minimizes F and 0 Â£ Vh. Therefore, we must have F-2(i>h) ~ 0 and F-i(ip) = 0, since otherwise F(iph) oo in the limit Â£ > 0, which contradicts (2.6). In combination with (2.5), we conclude that tfoCO = 0. . To prove (ii), by virtue of (i) we can restrict the minimization of F to the space ( OO OO wh := < wh Â£ C(D) : wh(z,n) = ^Â£"^(z) + //^V<5(z); v=0 V = 1 Wh(zi,fJ.) = 0 for fi > 0, Wh(zr,fj) = 0 for /z < 0 where tjv(z), 8v(z) Â£ IPr(Fh) are independent of Â£ for all v. A necessary condition for Â£ Wh to minimize F is that the first variation of F vanishes at rph for all admissible Wh Â£ Wh, that is, (Zi>h,Zwh) = (eq, Cwhj VwhEWh and V e > 0. (2.7) 15 For wh G Wh we have Cwh= \p-io++e[iiT)[+n26[->r 1182 +ar]o\ +s2 [ni2 + fi26'2 + /i63 + arn] + 0(e3). Therefore, (2.7) is equivalent to ZT 1 J Js $$vWo + Mo) + (?0<5i + dfidz + eli + e212 + 0(e3) zi -1 2r 1 H2qS[ + aqrjo dfidz + 0(e3), zx -1 (2.8) where zr 1 hJ J H2 (TlWi + Zl -1 Zr 1 h J J fJ-2 (V0V2 + M2) + (7o53 + M3) + (Mo + V^i) + (^3f?o + ^3<5i) Zl -1 +rjWi + V1&2 + od>[T)o + M'i + M2 + aMi] +H46[6[ + a2rjoTfo dfidz. Since (2.8) holds for all e > 0 and for all Wh G Wh, in particular for Wh = ij>k, it follows that ZT 1 Zr 0 = J J f!2 (rjon'o + 2r?D6i + Mi) dfidz = | J (rj'0 + 61) dz. Zl -1 Zl Thus, Finally, we prove (iii). Because of (ii), we can restrict minimization of F to the space Wh := {wh G Wh : rf'0(z) = 61(2)}. The choice w;t 6 Wh in (2.7) will zero out the 0(1) and 0(e) term in (2.8). Comparing the 0(e2) term on the left-hand and right-hand sides gives J g ^ivWi + Mi) + (M2 + 8262) + a (Mo + 77o<5i)] + r Mi + 2a2rjoTfo dz 0 dz (2.9) 16 for all qv 6 Wh with v > 0 and for all 6 G Wh with v > 1. From the choice 6[ = 0, ?j0 = 0, T)[ = rj[ and S2 = S2, we conclude that ZT J (jfi+Sz) dz = 0 => rj[ = -S2. (2.10) ZJ Substituting (2.10) into (2.9) results in J (^i^o + rjo^Cj + ^i6i + Z^VoVo dz = J ^qS[ + 2aqrjo dz. z\ Z\ Choosing <5i = 0, then integration by parts leads to %r Zr j a6irj'0 + 2a2T)Q-qQ dz = 2a J qr]o dz, Zl Z\ which with (ii) and after division by 2a becomes zr z r J \vWo + Wo dz = J qr)o dz. (2.11) Zl Z\ Because of the choice wj, G Wh, equation (2.11) holds for all 770 G Uh. Q One major implication of Theorem 2.1 is that, when tj(z) and <5(z) are continuous piecewise linear functions, (ii) can only be fulfilled if rjo is a linear function. Otherwise, Si = t?0 is a step function, which would not be continuous. Taking the boundary conditions into account, it follows that rjo = 0. Therefore, Uh = {0}, so that (iii) is a vacant statement and does not contradict the fact that rjo = 0 is a solution. Consequently, in the diffusion limit e 0 the discrete minimizer i>h converges to iph = 0, independent of the choice of the right hand side q. This shows that the Least-Squares Finite-Element discretization of (1.6) with linear basis elements in space does not give a correct diffusion limit approximation, except in the case q = 0. For a different way of proving this result, we refer to (Manteuffel and Ressel [38]). On the other hand, for piecewise polynomial basis functions of degree r > 1, con- dition (ii) does not restrict rjo to a linear function. Therefore, Uh contains also nontrivial functions, so that (iii) implies that rjo is a Galerkin approximation of the diffusion equation + = q. Thus, the Least-Squares Finite-Element discretization with piecewise poly- nomials of degree r > 1 yields a correct discrete diffusion limit solution. However, numerical results for a discretization in space by piecewise quadratic basis functions show that applying a scaling transformation (introduced in the next section) prior to the discretization enhances the accuracy. 2.2 Scaling Transformation In this section, we introduce a scaling transformation that is applied to the transport operator prior to the Least-Squares discretization. This scaling transformation plays a key role in this thesis, since it guarantees the accuracy of the Least-Squares discretization in 17 diffusive regimes even for simple Finite-Element spaces, such as spaces using continuous piecewise linear elements in space (see Section 2.5). To motivate the scaling transformation we introduce the moment representation of the flux. Let Pi(fj) denote the 1-th Legendre polynomial. The normalized Legendre polynomials pi(fi) := y/2l + 1 Pi(p) form an orthonormal basis of L2{[ 1,1]): l \ JPk(p)pi(p)dfi = 6ki, (2.12) -l where Ski denotes Kronecker delta, i.e, Ski 1 for k = l and Ski = 0 otherwise. Assuming that il>{z,n) 6 L2([1,1]) for all z 6 [z;, zr], then ij) has the following expansion (moment representation) in angle: CO HZP) = Mz)pi(p)< (2-13) 1=0 where the Fourier coefficients i{z), which are called moments in neutron transport theory, are given by l j'l>{z,p)pi(p)dP- (2.14) -l We see directly that the projection operator P, defined in (1.7), is a projection onto zeor moments (Pip = o), the operator Pfi is a projection onto first moments (Ppip = and the operator (IP) is a projection onto moment one and all other higher moments. With the concept of the moment representation, the diffusion expansion (1.13) leads to the implication that, in diffusive regimes, only moment zero and one are the important components of the solution. Because of Ceas Lemma (1.24), the solution of the Least-Squares discretization can be viewed as the best approximation to the exact solution in the discrete space Vh with respect to the norm \/a(-, ) := < Â£,Â£ >. However, the different terms in the operator Â£, as defined in (1.11), are unbalanced (there are 0( j), 0(1) and 0(e) terms), so that different components of the approximation error are weighted differently in \/a(-, ). The leading term of Â£ is ~(I P), which means that the error in the higher moments is weighted in this norm very strongly in diffusive regimes (very small e), even though this part is not important according to the diffusion expansion (1.13). On the contrary, the error in moment zero, which is the important part in diffusive regimes, is hardly measured in the norm y/d(-, ), since it is weighted by e. The basic idea is, therefore, to scale equation (1.11), thus changing the weighting in the norm used in the Least-Squares discretization to determine the best approximation to the exact solution in the discrete space. Define for r 6 JR+ the following scaling transformation and its inverse: S~P + t(I-P), S-1 =P + ^(I-P). (2.15) After applying the scaling transformation S from the left and dividing by e, equation (1.11) becomes 1 ~ 1 (9?/} 7* Cil> := -SCi> = -Sii-Â£ + -^(I-P)4> + aPi> = qs, (2.16) 18 where q, Sq and l ^ dip \ dip t dip = -Pn-g- + J P)^. e dz Â£ dz Â£ dz Clearly, choosing r = 0(e) will increase the weight for moment zero and reduce the weights for the higher moments. Equation (2.16) can be balanced further by a scaling transformation from the right. Let the domain of operator C in (2.16) be the Hilbert space V. Then we define the space V by V := S'-1 V, so that ^ _ v = S_1v for all Â£V and Sv = v for all v E V. \ J Scaling (2.16) also from the right results in ^ 1 f)tb 7^ CSS~liP = CSiP = -SiiS^f- + -r{I P)i> + aPiP = qs, Â£ OZ Â£* (2.18) where -SfiS = -(r t2)(Ph + fiP) + e Â£ For r = 0(e) we have f E 0(1), so that in (2.18) the derivative of moment zero and one and the moments themselves are weighted equally. Moreover, we point out that the double-scaled operator CS can be bounded independent of Â£. In the Least-Squares context, the additional scaling from the right can be avoided, since min {CSiP qs, CSip qs) <=> min {Cip qs,Cip qs), (2-19) which will simplify the boundary conditions and so also the computations. Further, for slab geometry, because of transformation (2.1) we may assume without loss of generality that at and parameter Â£ are constant in space. However, for higher dimensional problems, this cannot be established, so that Â£ = e(r). For inhomogeneous material, e(r) is in general discontinuous, so that the scaling parameter r, which was chosen to be 0(e), would be discontinuous. To perform the scaling would then require to prescribe jump conditions in the scaled solution v across material interfaces. Therefore, we use the additional scaling from the right only as motivation for the choice of the scaling transformation and as a tool in the theory in Section 2.4, where we exploit the nice form of the double scaled operator (2.18). For another way of motivating the scaling by way of the moment equations, we refer the reader to Manteuffel and Ressel [38]. As outlined in Section 1.4, a necessary condition for ip E V to be a minimizer of the Least-Squares functional (2.19) is that the first variation vanishes at ip, which results in the problem: find ip EV such that a(ip,v) := (Cip,Cv) = (qs,Cv) Vu G V. (2.20) For a discretization of problem (2.20), the bilinear form a(-, ) is restricted to a finite di- mensional subspace Vh C V. In the remaining of this chapter, we analyze the error of this discretization for various subspaces Vh. 19 2.3 Error Bounds for Nondiffusive Regimes In this section we establish bounds in an unsealed norm for the discretization error of the Least-Squares discretization. However, in this norm it is not possible to prove V- ellipticity and continuity of the bilinear form (2.20) with constants independent of parameters e and a. In diffusive regimes, where e is very small, these bounds blow up and are therefore useless. Nevertheless, the bounds for diffusive regimes that are derived in Section 2.5 are only valid for e < l/\/3, so that the bounds in this section can be used to cover the range Â£ > 1/v^- As outlined in Section 1.4, the first step on the way to bounding the error is to prove V'-ellipticity and continuity of the bilinear form a(-,-) in some norm. From the view of standard elliptic boundary value problems, the choice V = ff1([z/) zr]) x L2([ 1,1]) (Adams [1]) with the norm l?,o //(Â£) +v3ilMlz seems natural. However, it is easy to see that the bilinear form a(-, ) cannot be bounded from below in this norm. Let Vk := \/2 sin(&7rz) B(/x) with B(ji) := /~3~ 6+li Y 26 S fÂ£tzJL V 26 6 for (i Â£ [6,0] for fi Â£ [0, <5] otherwise Then, for all k Â£ IN, we have Vk Â£ i?1([z/,zr]) x L2([1,1]) and ||fc||i,o = (kit)2 + 1. Some simple calculations show that , . 1 + r2 (kir)262 2r2 + 2s4a2 a[V, V) < ----5----------- + --------7----- Choosing 6 = then the bilinear form a(-, ) is bounded for all k while ^lim [|vjb||i,o = - Thus, there is no lower bound for a(-, ) in the norm 11-1]! 0. The next obvious choice is Ml|2:= dv lYz + V := {u Â£ C(D), v(zi,fi) = 0 for (i > 0, v(zr,fi) = 0 for fj, < 0}. (2.21) Closure here is with respect to the norm |j|-[[[, so that V is a Hilbert space. In the following, we bound the Least-Squares discretization error in norm (2.21) for various Finite-Element spaces. 2.3.1 Continuity and V-ellipticity Before we establish continuity and V-ellipticity of the bilinear form a(-, ), we sum- marize some simple properties of our operators. 20 Lemma 2.2 (Properties of P, S and fi-^) For all u,v G V, we have: (i) (Pu,v) = {u,Pv)i and {(I P)u, v) = {u, (I P)v); P2 P\ and (I P)2 = (I P). Thus P and (J P) are orthogonal projections; (ii) {Pu,v) = (Pu,Pv)\ and {(I P)u,v) = ((/ P)u,(I P)v}\ (0 IHI < -Sv e for scaling parameter r = e and e < 1. (iv) ||H|2> - \W -PMl) ; (v) (y,v)>0; l Proof. (i): Zf 1 1 (Pu, v) = f f Pu v dfidz f Pu f v d\idz Z\ -1 Z\ -1 Zf zr 1 = f 2Pu Pv dz f Pv f u d\idz Z l Zl -1 z r 1 = f f u- Pv d^dz = (u, Pv), Zl -1 and the second identity follows directly from the first. From the definition of P, it is obvious that P2 = P and, therefore, (I P)2 = (I P). (ii) : follows immediately from (i). (iii) : IHI2 = ||Pv||2 + ||(I P)u||2 < fHI^II2 + Wi1 ~ -PHI2 = Il'^SuH2, since e <1. (iv) : Zr 1 = l \\Pv\\2 + 2 J J n2Pv (I P)v dfi dz + MI P)v Zl -1 21 The mixed term can be bounded by the Holder inequality as follows: zr 1 J J n2Pv (I-P)vdfidz Zi -1 Zr 1 < J \Pv\ Jn[ji\(I P)v] dfidz *i -i Therefore |fi(I P)V>|2 d/i dz 1/2 IIHf > l \\Pv\\2 ~ ||P*|| ||/i(J P)*|| + MI P)v IM-IK/-p)|| 2 (v): Applying integration by parts with respect to z, we get Zr 1 1 Zr 1 J J /i^ -v dfidz = J (J, [v2(zr,n) v2(zi,n)] dji-J Jn^-vdfidz- Z\ 1 1 Z\ 1 Taking into account the boundary conditions for v E V, it follows that l fiJzV'} = \ J fI[v2(Zr>fJ')-v2(zhlJ')] dV -1 = ^|/fiv2(zr,fi) dfi- J^/iv2(zi,ii) dfij > 0. It now easily follows from the Cauchy-Schwarz inequality that the bilinear form a(-, ) is continuous in the norm |||'|||, since for any u,v G V Ku>)l = ||(Â£u,Â£v)| < l|Â£ll ll^ll < 1 + T du dz , T + Â£ a ii i + M 1 + T dv + t + sza HI < Ce lllulll IIIHII. Here Cc '= step. (ir)2 _j_ ^zP_gPj I ^ ancj we used the discrete Holder inequality in the last 22 We prove now V-ellipticity of the bilinear form a(-, ) when a ^ 0. Lemma 2.3 (V-ellipticity for a 0) Suppose a^O and let r = e^fa. Then there exists Ce > 0 such that, for all v G V, a(v,v) > Ce ||M||2, where Ce := min {^, a, a2}. Proof. We have a(v, v) = (jCVyjCrV) 1 dv 2 , dv e2 + a V~PT, + -t\\(I-P)vtf + a>$$Pv\\ (2.22) The second mixed term can be written using (ii) of Lemma 2.2 as () = ) - p) According to (v) of Lemma 2.2, the first term here is always positive and the second term cancels with the first mixed term in (2.22), so that a(v,v) = 1 dv 2 . dv e* + a + ^\$$I-P)v\\2 + a2\\Pv\\ > Ce IIM with Ce := min { 77, a, a2}, which proves the lemma. For the more difficult task of establishing the V-ellipticity of the bilinear form a(-, ) when a = 0, we need the following Poincare-Friedrichs inequality. Lemma 2.4 (Poincare-Friedrichs Inequality) For any v G V, we have \\flV < Proof. We have f dv(s,n) r torf- Zl < f dv(s, fi) ~Jtai-0 . Z (2.23) 23 < < z 1 zi z r I 9v(s,n) ds dv(s,n) ds ds for fj, > 0 ds for fx < 0 Zt / 2>r < J dz < (zr ~ Zi)1/2 | J |/i dv dz 1/2 dz Taking into account that assumption (zr z/) = 1 implies IIHI2 < we obtain the lemma. We are now in a position to establish Lemma 2.5 (V'-ellipticity for a = 0) Suppose that a = 0 and 0 < e < 1, and let dv T = yf' 2 + 52 ^ = 7 + V1 ~ P)/*Â£ + 72 (I-P> + Pv- Then there exists Ce > 0 such that, for any v G V, a(v>v) > Ce [|MI|2 i where Ce = ^54. Proof. Recall that 1 dv -Pp-z~ Â£ dz Â£ Because of (i) in Lemma 2.2, we have a(v,v) = {Â£v,Â£ v) 1 / dv dv\ r2 / dv 2r2 / f)v \ <(' P>> V ~ P + IF {(P~ P)^ V ~ P>) Analyzing the mixed term and using (i) of Lemma 2.2, we see that [v-^TzV- F}") = ({I~ p)pf?) = (''I' ) (P"S'p 24 The first term is always positive according to (v) of Lemma 2.2. Consider the following arithmetic-geometric inequality: for any rj E 1R+ and for any a, be 1R, 2ab < -qa2 + We can thus bound the second term according to P^pv)< dv P^z ll-Pwll 2 dv ^Tz Therefore, the bilinear form a(-, ) can be bounded from below by a(v, v) > dv 2 T2 , a dv ^z H2 2 U-F>S +Â£ll (I ~ PM2 ~ ^\\Pv\\2 . Defining 6 := [|Pu||2 l|2 dv I T;=Jtei /Sir' SO that q ( T)_ Wit the above bound thus simplifies to a(v,v) > Ci dv % + ^2 |M|2 with Ci = ^2 {j--Jt,7 + t2^1~7^) ft ?(Â£>-*>-Â£), (2.24) By proper choice of 77, we now need only establish that Ci and C2 are positive. Unfortunately, for large enough 6, C2 will be negative, so we will need in this case to readjust the terms in (2.24), which we do by way of the Poincare-Friedrichs inequality. Case 1: S > yÂ§: From the Poincare-Friedrichs inequality (2.23) and (iii) of Lemma 2.2, we conclude that dv Tz >\\H\2>[^\\Pv\\-MI-P)v\ Since fi E [1,1], then clearly ||/i(J P)v[| < ||(/ P)u||. Therefore, ||P|| ||ai(J PHI > ^ ||P|| ||(I P)VII > 0, where the last inequality follows from the assumption 8 > y| > | since (2.25) i||Pu||2>||(J-PH|2^35>(l-^) >-i 25 From (2.25), we get -J= \\Pv\\ MI pm) 2 > ||Pt,|| II(I P)vI It then follows that IHJlf Thus, We now use (2.26) to rewrite (2.24) as a(v, v) >[Ci-j >|C1 + 5? dv 1 dz dv "Tz + 2e2 dv Tz + c2 ihi + l^.+ C2)|M|2 V26e2 Choosing 7] = -j- and using the fact that 6 < 1 results in T^ 1 / +Ci = 26s2 and = ?(?(1-)+,(s-b))^^> c-Â£*-?(i(1-tJ'(1+S)) + tJ >yi since t 1/^2 + fÂ§, so r2 (l + <1. Poo/s O* X / i^ Plinrvoinrr m ---- 0/1 Case 2: <5 < 1|: Choosing 77 = 24s, for Cl and C2 in (2.24) we obtain that and ' r2 / 25 \ r2 / 12 25\ r2 C2 e4 ^24 j e4 V1 13 24) 26s4 > Ci = \ (7 (1 24t2) + r2 (1 7)) > ^ > - since 24r2 = 7^2 < 1 for Â£ < y/ff (2.26) 26 Thus, altogether we have a(v, v) > Ct 2 with which completes the proof. From continuity and V-ellipticity of the bilinear form a(-, ) it follows directly from the Lax-Milgram Lemma (see Section 1.4) that problem (2.20) and all of its discretizations are well posed. The next step is to obtain discretization error bounds for a variety of discrete subspaces Vh, which is done in the next subsection. 2.3.2 Error Bounds As outlined in the introduction, continuity and P-ellipticity of the bilinear form a(-, ) lead directly to Ceas Lemma (1.24). Therefore, bounding the discretization error ||\ij) VViIII is reduced to the problem of bounding min |||?/> /i|||, which is a problem of vK&Vh approximation theory and depends on the choice of the finite-dimensional space Vh. Here we consider two main classes of discrete spaces Vh. The first consists of spaces with functions that can be expanded into the first N normalized Legendre polynomials with respect to the direction angle p and are piecewise polynomials of degree r in z on a partition Th of the slab [z\, zr\. This choice of the finite dimensional space Vh corresponds to discretization by a spectral method in angle p and a Finite-Element discretization in space. In transport theory the spectral discretization in angle with the first N Legendre Polynomials as basis functions is also called a -P/v-i discretization in angle. For any f(z) Â£ Hm([zi,zr]), with 1 < m < r+ 1, let HZf{z) denote the interpolant of f(z) by piecewise polynomials of degree r > 1 on a partition of [z\, zr\. It then can be shown (Johnson [25, p. 91]) that \\f(z) nj(z)\\LH[zitZr]) < Chm Wf^\{z)\\L^[zuzA) [f(z)-nzf(z)] < Ch! Â£2([*.,*r]) 1 /(m)(z) (2.27) where h is the maximum cell-width of the partition and the constant C is independent of h and /. Further, for any g(z, fi) Â£ Hm{[zi,zr]) x H2([ 1,1]) we define JV--1 n;vÂ£f(z,M) := XI with Mz) r=o l \ j 9(z,p)pi(p)dv -i (2.28) as the truncated expansion of g into the first N normalized Legendre Polynomials pi(fi) (see (2.12) in Section 2.2). Note that the normalized Legendre-polynomials form an orthogonal basis of L2([ 1,1]) and are the eigenfunctions of the Sturm-Liouville operator (Gottlieb and Orszag [21, p. 37]) Â£sPi(n) ~ A. \(i dP'(p) dp _ ' dp = 1(1+ l)pi(p). (2.29) 27 Then the error of the truncated expansion can be bounded as follows Lemma 2.6 (Truncated expansion into Legendre polynomials) For r > 0, let g(z,[i) 6 Hr([zj,zT]) x H2([ 1,1]) and let 11// be defined as in (2.28). Then have: IIIM < (ii) <_______ - '('+1) (iii) For any m Vro < r and Vz G [zi, zr]; dmg C dmg dzm 11jv dzm ~ N s dzm (2.30) with C independent of g and N. Proof. (i): II// is orthogonal projection with respect to the inner product of L2([1,1]), so ||n/n7||i2([_li;l]) < IMIL2([_u]), and hence ||IIjvff|| < ||ff||. (ii): By. definition (2.29) and integration by parts, we have jJrHz) =lf ~ 21(1 + 1) / Cs dz P,(M) dp ~ ^21(1 Therefore, < /(/+!) Cs + 1) dmg Cs dmg dzm Â£2([-Ul) (2.31) Sz (iii): Since the Legendre Polynomials are an orthogonal basis, from (2.31) we obtain dmg dmg - II// dz dzn = 2Â£ |^m)WI2< i2([-l.l]) l=N Cs dmg dzr Â£2([-Ml) 1^2 For / > 1 we have j < so that the sum can be bounded by OO o - | . OO ^ TO ^ y I 4 4 Therefore with C = yjÂ§, which proves the lemma. dmg <9mff ^ C dmg <9zm N dzm ~ N Ls dzm 28 Theorem 2.7 (Finite-Element in space, spectral discretization in angle) Suppose that 7ft is a partition of the slab [27, zr\ with maximum mesh size h. Let V be given as defined in (2.21). Define {JV1 vh e c(Dy, vh = J2 (*) e JPtVh) for / = o,..., n -1 1=0 vh(z,,n) = 0 for n > 0, Vh(zr,fj,) = 0 for n < 0 where 2Pr(7/l) denotes the space of piecewise polynomials of degree < r on the partition 7ft. Suppose 1 < m < r + 1 and let ij> G Vn (Hm([zi, zT]) x H2([-1,1])) be the solution of (2.20) and iph Â£ Vh be the solution of (2.20) restricted to Vh. Then VIII \\Csi>\\h0 + C2h m1 dmip dz" Proof. From Ceas Lemma (1.24), we have WH-MW < min VhÂ£Vh HIV1 vh < |||V> Hivn^lll Now note that |||?;||| < Q. Therefore, by (i) of Lemma2.6, (2.27) and (2.30), we conclude III n^n2v>||| < I]i> nJvV,llij0 + - n^V,)lli,0 which proves the theorem. The second main class of finite-dimensional spaces considered here are formed by functions that are piecewise polynomials in space z as well as in angle fi. This choice corresponds to a Finite-Element discretization in both space and angle with rectangular elements. Suppose that 7ft is a partition of the computational domain D = [27,27.] x [1,1] into rectangles T = [zj,zj+1] x [//, fJ,v+i] of maximum diameter h. To be able to handle the boundary conditions properly, we assume in addition that (27,0) and (zr, 0) are nodes of the triangulatiori 7ft. By 7ft we define the discrete space: dmip dzm yh vheC(D); *ft|T= 0 (2.32) Vh{zi,n) = 0 for n> 0, Vh(zr,fi) = 0 for fi < 0 29 For all v Â£ V, let H/,u Â£ Vh denote an interpolant1 of v with respect to the partition 7*. It can be proved (Ciarlet [16, Theorem 16.2]) that, for v Â£ V n7Tr^"1(Â£)) the following bound for the interpolation error holds: 11^ ~ < Chr+1~m \v\r+1, (2.33) where 0 < m < k + 1 and | \h-+'l(d) is the semi-norm of Hk+1(D) (Adams [1]). Combining Cea's Lemma and (2.33), we get Theorem 2.8 (Finite-Elements in space and angle) Let Vh, h be given as defined above. Suppose ip Â£ V D Hk+1(D) is the solution of (2.20) and let iph Â£ Vh be the solution of (2.20) restricted to Vh defined in (2.32). Then we have: \U-M\ < \f^chr\ip\Hr+1(D). Proof. By Ceas Lemma, we need only to bound |||V< II/1i/,|||- Note that, for all v Â£ V H Hr+1(D), IIMII < Hwllj 0 < ||v||ffl(1J). Thus, using (2.33) with m = 1, it follows that |||-0 Hft'0111 < chr \i>\Hr+^D), which proves the theorem. We point out that the error bounds in Theorem 2.7 and Theorem 2.8 depend on the ratio In the V-ellipticity bounds in Lemma 2.3 and Lemma 2.5, the scaling parameter r was chosen to be 0(e). Therefore, when e is small, Ce is 0(a) when a: / 0, while Ce is 0(1) when a = 0. In addition, for r = O(e), the continuity constant Cc is O(j-), so we have = O(j-), which blows up for diffusive regimes, where e is very small. However, numerical results show that the Least-Squares discretization of the scaled transport equation stays accurate in diffusive regimes. Thus, we conclude that the bounds, derived in this section, are not sharp enough to reflect the accuracy of the Least-Squares discretization in diffusive regimes. In order to obtain error bounds that do not blow up in diffusive regimes, it is essential to prove continuity and F-ellipticity of the bilinear form a(-, ) with constants independent of parameters e and a. This is done in the next section with respect to a scaled norm. 2.4 Continuity and V-ellipticity with respect to a scaled norm In this section, which is the central part of this thesis, we prove continuity and V-ellipticity of the form a(-, ) in (2.20) with constants independent of parameters e and a. This is the foundation for the bounds in Section 2.5 of the Least-Squares discretization error that do not blow up for diffusive regimes. Throughout this section, we assume that t = e and a < 1. In order to obtain continuity and V-ellipticity with constants independent of e and a, we use a scaled norm. To motivate its choice, we look at the double-scaled (from left and right) 1For r > 2, there axe many different interpolants, depending on the choice of the support abscissas and support ordinates on the rectangle, which are not specified here. For an overview of commonly used interpolants for rectangles, we refer the reader to Ciarlet [16, p. 129]. 30 transport operator (2.18). Let V denote the domain of the single-scaled (only from the left) transport operator (2.16) and V = S 1V the domain of the double-scaled transport operator (2.18). Defining Q := jSfiS = (l-e)(Pfi + fiP) + Â£fil, (2-34) we see that the norm 2^ + IHI2 (2.35) for v Â£ V would be a natural choice for bounding the double scaled bilinear form a(u,v) := (Â£Su,Â£Sv) . (2.36) However, because of the reasons mentioned in Section 2.2, it is desired to use the single- scaled transport operator for the computations. Therefore, using the relation v = 5_1 V, we derive from (2.35) the following norm for v Â£ V: II = jSfiSS- -i dv dz ii 1 dv -Sfi-K- Â£ OZ 2 + = 1 dv -Sfi-K- Â£ OZ 2 + + lls-M Pv+ -(I-P)v Â£ (I-P)v + \\Pv\\2 =: \v We define (2.37) V := {v Â£ C(D); v(zi,fj,) = 0 for fi > 0; v(zrifi) = 0 for (i < 0}, (2.38) where closure is with respect to the norm |] ||y, so that V is a Hilbert space with respect to the inner product {u,v)v := /^sfi^,^Sfi^\ + /^(I-P)u,^(I-P)v\ + (Pu,Pv). At a first view, the norm ll-llv also seems to be useless for error bounds when e > 0. However, suppose we could show that the error of discretization is bounded by \W ~h\\v ,qs,h,N), with lim qst h, N) - 0. iV oo, h+ 0 Then, in particular, every part of the norm ||^i i>h\\y is bounded by C, so that ||(7 P)(ip iph)\\ < eC, for example, which means that the error in the higher moments is decreasing with e. First, we prove continuity of the bilinear from a(-, ) in the norm H'lly. From the Cauchy-Schwarz inequality, it follows directly that, for any u,v Â£ V, |a(u,u)| = |(Â£u,Â£u)| < ||Â£u|| H-Ci'H . 31 Employing the discrete Holder inequality and using the assumption a < 1, we obtain IIjCuII < Isc- e *dz + -(I-P)u Â£ + PM 1 du 2 1, -Sfi-K- e dz + + ll-Pu; \ 1/2 I2 =V3 < V3 Thus, for all u, v 6 V, |a(u,v)| < Cc ||||v |M|V with Cc = 3. To prove V-ellipticity of the bilinear from a(-, ), we exploit the convenient form of the double-scaled transport operator and prove first that the double-scaled bilinear from a(-, ) in (2.36) is V-elliptic. The V-ellipticity of the bilinear from a(-, ) then follows easily as in Corollary 2.12. In order to prove V-ellipticity of the bilinear from a(-, ), we need the following lemmas. Lemma 2.9 For all u, v 6 V, v Â£ V and e. < 1, we have (i) ^ll^ll e||(J p)v\\ < IMI; (ii) {QTz'*) -0; (iii) (Poincare-Friedrichs inequality) IIHI < Proof. dv 1 dv dv < ^Yz < -SfiS Â£ dz QY (2.39) (i): Since \\fi(I P)w|| < ||(I P)v[| and ZT 1 Zl -1 l J J /x2 (Pv)2 dfidz = J (Pv)2 J /i2 dfidz = J (Pv) 1 Zl dz z r 1 \j J (Pv)2 dtidz=\\\Pv\\2, Zl -1 then iihi = iia*^+- p)]v ii > ii^ii c\w p)* i: >^npii-eii(/-p)% (ii): From (i) in Lemma 2.2, it follows that (Su,v) = (u,Sv) Vu, v E V. Therefore, using (2.17) leads to | (f?s) = (H5!?') =; (^st) = l ('Â£) where the last inequality follows from (v) of Lemma 2.2. 32 (iii): From the Poincare-Friedrichs inequality (2.23) proved in Lemma 2.4, we have ||^|| < llpfell- Using (iii) of Lemma 2.2 and the relation (2.17) results in dv 1 dv 1 dv dv "T* < -SV7T e dz -SfiS=- e az The following technical lemma is tedious but simplifies the proof of the major result, Theorem 2.11. Lemma 2.10 Suppose 0 < e < Then, for any 6 Â£ [0,1], there exists 6 > 0 such that H(b, 5) := | y^6 -(1+ Â£)&)' + 46(1-5)+ (l J) b + 6 j < 0.988, where s := j^1/2 e\/3(l 6)^2j In particular, for 5 < 0.875, we can choose 6 = 0. Proof. For 6 < 0.875, we choose 6=0 and get 77(0,6) = i |\/46 362 + 6} < 77(0,0.875) < 0.986, since 77(0,6) is monotone increasing for 6 Â£ [0,1]. For 6 > 0.875, using the assumption e-y/3 < 1, we have s = [61/2 eV3(l 6)1/2]2 > [61/2 (1 6)1/2]2 = 1 2^6(1 6) =: (3. Suppose we restrict the choice of 6 to 6 < |6. It then follows that < (- (i+1) <) < (- (*+1) o s (* C1+f) 0 since s < 1. From (2.40), we conclude that (s-(1+l)t)"s (i-(i + f)t)2. Therefore, (2.40) H(b, 6) < U J (s (i + Â£) &) + 46(1 6) + (l f'] + > =: 77(6,6). Simple calculus shows that 36 _ (3-/?) 36(1-6) 3 (3 + /?) (3 + /3)V P ~4 33 minimizes H and that b* > 0 if 6 > 0.875. After tedious but straightforward manipulation, we have that H(b*, 6) attains its maximum at 5* 0.893 and that H(b*, 6*) < 0.988. We are now ready to prove the central result of this section. Theorem 2.11 (F-ellipticity of a(-, )) Let a(-, ) and || ||~ be given as in (2.36) and (2.35). Suppose that 0 < a < 1, 0 < Â£ < ^=. Then there exists a constant Ce > 0 such that, for all a(v, v) dv Q-q^ + aPv + (I P)v >Ce + M2)=Ce (2.41) where Ce = 0.012, which is independent of e and a. Proof. We have d(v, v) = dv Q + aPv + (I P)v oz + a2||PS||2 + |](L-i>||2 +2 (Â£)+:2 (^,-p)") + a2\\Pvf + \\(I-P)v\\ +2a ( Qyz, v\ + 2(1 a) /q||, (I P)v (2.42) For the last term, we may write for any d G [0,1], by using (ii) of Lemma 2.2 and (ii) of Lemma 2.9: d (q^, (I P)v} + (1 -d) (Q^z, (I P)v} = d(Qtd)'d (Qt *>) + (1 d) (Qt V ~ P*) > -d (pQyz, Pv} + (1 d) ({I P)Q^z, (I P)v} . Substituting this into (2.42) and bounding the fourth term in (2.42) by (ii) of Lemma 2.9, 34 we get Z(v,v) > ||g|f|| +a*\\Pv\\* + \\(I-P)v\\* -2(1 a)d\(PQÂ§,Pv) | 2(1 a)(l i) |((J P)Qf, (I P)*>| which can be reduced by setting a = 0 to a(v,v) > ||Qf||2 + ||(7_P)l;||2 -2d | (PQÂ§, Pv) | 2(1 d) |((7 P)QÂ§, (I P)v) |. Defining <:=!Â£Â¥, so that (1 ) = IIMC, (2-43) 12 npqj?ii solhat fl T)_ii(f-p)osir Htf llelfir ' and using Cauchy-Schwarz inequality, we conclude from (2.43) that a(v, v) > Q dv dz + (1 5)||t)||2 2dV6y/j -2(i-d)V(T^)V(i^) tdv Tz (2.44) To maximize the lower bound in (2.44), we divide the region (5,7) E [0,1] x [0,1] into two triangles and choose d as follows: :-{i for 5 + 7 < 1 for 5 + 7 > 1 ' Next, we consider these two cases separately. Case 1: 5 + 7 < l,d = 1: For any 77 > 0 and any u,v and any norm || ||, the arithmetic-geometric mean inequality 2IMHMI < 7lMI2 +;HMI2 holds. We thus have a(v, v) > (1 777) 2 + 1 (2.45) It remains to choose 77 such that the terms (1 777) and (1 5 ^) in (2.45) can be bounded by a positive constant from below for all possible 7,5 with 7 + 5 < 1. For 5 < 0.5, we choose 77 so that (1 77?) = (l 5 0 , 35 which yields 77 = 6 + y/6*~+467 Applying Lemma 2.10, since 7 + 8 < 1 and therefore, 7 < 1 <5, then we have 71 = \{6 + V^+467} < \ + y/P + 45(1 5)} = H{ 0,8) < 0.988. Thus, from (2.45), the V-ellipticity of a(-, ) directly follows with Ce > 0.012. On the other hand, if 8 > 0.5, the second term in (2.45) can become negative. To keep it positive, we then rewrite (2.45) for any 6 6 [0,1] as follows a(v,v) > (1 b yr]) Q +b Q + (1 6 - ^dv 2 _dv Q9! + 6 % Since 8 > 0.5 and e < l/\/3 we have 0 < ^^=V6 ey/1 6^. We can now use the Poincare- Friedrichs inequality (2.39) of Lemma 2.9 and inequality (i) of Lemma 2.9 to bound the second term by o _ 2 V6-ey/0^S) ||i;|p dv 2 r > imi2 > which results in a(v, H) > (1 b 777) + (- - b 5 8+-s----- 3 77 where Again, we choose 77 so that s := (v^-ev/Ml-^))2- (1 6 777) = (\ 8 + ^ (2.46) (2.47) which yields ___________________ - (6 + |s 8) + J(b+ fs 6)2 + 467 V = 27 ' Next, to attain a positive constant in the lower bound in (2.46), we need to show that for all possible 8,7 with 8 + 7 < 1, 8 > 0.5, a positive b can be selected so that 1 Gi{b, 8,7) 6 + 777 6 + - 6^ + 467 + fb + 8 - } < 1. Since Gi(6,8,7) < Gi(6,8,1 6) for 8 + 7 < 1, it then is sufficient to prove V6e [0.5,1] 36 >0 : Gi(6,8,1 8) < C* < 1. But this follows immediately from Lemma 2.10 with G* = 0.988 since Gi(6,8,16) = H(6,6). The V-ellipticity of a(-, ) then follows directly from (2.46) with Ge > 1 0.988 = 0.012. 36 Case 2: 5 + 7 > 1: Setting d = 0 in (2.44) and preceding as in case 1 results in 2 d(v,v)> (1-(1-7)77) dv ' dz +1-5- 1-6 For 6 < 0.5, we choose so that S+^+4(l-5)(l-7) V 2(1-7) Using Lemma 2.10, since 6 + 7 > 1, and therefore, 5 > 1 7, we then have 1 (2.48) (1 (1 7)77) = ^1 6 - - 7 > 1, and therefore, <5 > 1 7, w< (1 -7)V = \ |\A2 + 4(1 5)(1 7) + 5} < | {y<52 + 4(l-5)5 + 5} = H(0,6) < 0.988. The Vr-ellipticity of a(-, ) then follows with Ce = 1 0.988 = 0.012. On the other hand, for 6 > 0.5, we introduce, as in case 1, a parameter b [0,1] and use the Poincare-Friedrichs inequality (2.39) to conclude from (2.48) that a{v,v) > (1 b (1 7)77) S+-s- 1-6 f] (2.49) with s as defined in (2.47). Again, we first choose 77, so that (1 6 (1 7)77) = (l 6 + Is - which yields O + Is s) + \/(&+!s-<5) +4(1 <5)(1 7) V~ 2(1-7) In order to attain a positive constant in the lower bound (2.49), we need to show that, for all possible 6,7 with 6 + 7 > 1, 6 > 0.5, a positive 6 can be selected so that, G2(b,6,j) := 6+ (1-7)77 1 2 b+-s-5 2 + 4(l-)(l-7) + i + -3* < C* < 1. Since G^b, 6,7) < G2(b, 6,15) = H(b, 6) for 5+7 > 1, this follows directly from Lemma 2.10 with C* 0.988. Finally, from (2.49) with Ce := 1 C* = 0.012 the U-ellipticity of a(-, ) follows, which proves the theorem. From the U-ellipticity of the bilinear form a(-, ) the U-ellipticity of the bilinear form a(-, ) can be proved as follows. Corollary 2.12 ( U-ellipticity of a(-, ) ) Let a(-, ) and H-Hy be given as in (2.20) and (2.37). Assume that 0 < a < 1, 0<Â£< Then there exists a constant Ce > 0 such that, for all dGU, a(v,v) > Ce \\v\\2v, (2.50) 37 where C& = 0.012, which is independent of a and e. Proof. By the definition of the norm ||-||K in (2.37) and the relation (2.17) we have = ||i)||p. Therefore, using (2.41) in Theorem 2.11 we obtain for any v 6 V a(v, v) Â£v Â£v dfidz Â£Sv Â£Sv dfidz = a(v,v) > Ce KM 2 V > which proves the corollary 2.5 Error Bounds for Diffusive Regimes Using continuity and V-ellipticity of the bilinear from a(-, ) in the norm ||-||v with constants independent of a and e, in the following section we establish discretization error bounds that do not blow up in the limit e 0. We use the same discrete spaces introduced in Section 2.3.2. We first consider discrete spaces with functions that can be expanded into the first N normalized Legendre polynomials with respect to the direction angle fi (Pn-i discretiza- tion in angle) and are piecewise polynomials of degree < r in z on a partition 7), of the slab. To combine Ceas Lemma (1.24) with the interpolation error bounds in (2.27) and (2.30) to obtain an error bound for this class of discrete spaces, we need the following lemma. Lemma 2.13 (Bound for commutator [IIjvÂ£ Til at]) Let Â£ be the transport operator as defined in (2.16), Â£s the Sturm-Liouville operator as defined in (2.29) and II/y the projection operator as defined in (2.28). Suppose IV > 2, m > 0, and v Â£ V fi Hm+3(D). Then, there exists a constant C > 0 independent of e and a such that [BjvT TII/v] dmv dzm c dm+1v - N2 Lsdzm+1 (2.51) Proof. Recall that and that has the moment expansion (see (2.13) in Section 2.2) dmv dzm = Â£ Am\z)pi{p) with = \ ( 9 ;=0 Ti Note that and ftin /i , v v n n p = <^nm p n jf " dzm 0 dzm N-1 = n* 4, = <Â£im) = Am)pi = p 1=0 38 Therefore, [njv^-^Djv] = UN fi-^- fiUN OZ Oz Using the relation (see Chapter 4) P Pi (p) = k-1 p(_ i (fi) + hi pl+1 (fi), with h - l + 1 v/4(/+ 1)2-1 and p~i(p) = 0, we see that d dmv UHIMTzd^ =IIiT Pi+i Lf=o (=0 AT JV-2 1=0 1=0 On the other hand, d dmv dz dzm =/-x;V+v 1=0 = ^2 <{>\m+1)bi-iPi-i 1=0 N-1 + Â£#! 1=0 (m+l) , Thus, in combination with (2.52) we have dmv [n^-Â£ Â£Hjv] = (j)^+^ 6jv_i pn-i ~ ^n-i Pn = bN-i (4>p?+1'*pn-i - Now we notice that, for any integers k, l > 0, / (m+l) (2.52) (2.53) ZT 1 Zr J (<^;m+1)(z)) j p\{p) dpdz = 2 j (<^m+1)(z)) dz Zi -1 z I t) (m+l) < W +1)]2 Cs 3m+1v 3zm+! where the last inequality follows from (ii) of Lemma 2.6. Therefore, (2.53) can be bounded 39 as follows: [IIjvÂ£ Â£II^r] dmv dz" N + V4AT2 1 \N(N 1) N(N + 1) Â£s dm+1v dzm+1 JV2- 1 Â£s dm+1v dzm+1 < 1 8 1 Â£s dm+1ip dz*+' y/3 3 N2 since N^_1 < O x 2, < |, which is valid for N > 2. We are now ready to prove the following error bound. Theorem 2.14 (Finite-Element in space, spectral discretization in angle) Suppose that Th is a partition of the slab [zj, zr\ with maximum mesh size h. Let V be given as defined in (2.38). Define Nl V ft ___________ Vh e cm vh=Y, WnOO; (*) e iPr(Th) for / = o,..., n -1; [=0 Vh(zi,fi) = 0 for // > 0 Vh(zr,fi) = 0 for /i < 0 > , where TPr{Th) denotes the space of piecewise polynomials of degree < r on the partition Th- Suppose 0 < e < 0 < a.< 1 and 1 < m < r + 1. Let ip E VflHm+3(D) be the solution of (2.20) with right-hand side qs E Hm+2(D). Let iph G Vh be the solution of (2.20) restricted to Vh. Then \\lp ihWv 2 + l(I-P)(lP-lPh) +\m-Tph)\\2 1/2 < 1 vc,; Ciu r ,ca C3hm dmg, dzm hm N2 Cs dm+1ip dzm+l > with Ci,C2, C3, C4 independent of a and e. In particular, (2.54) Proof. d(ip iph) dz Using Ceas Lemma (1.24) < eeh, \\(I P)(tp iph)\\ < eeh, < eft, \\P(ip iph)\\ < eft. and the U-ellipticity of the bilinear form a(-, ), from 40 Corollary 2.12 we conclude that HV-V/illv < ~^rVaii> iph) < -7= min \/a{ip vh,ip vh) VOe vOe vheVh < ~^r^/a(^ ~ ^z^-Nip, ip TlzllNip) = -J=z \\Â£(ip n^njvVOII (2.55) < -j= {\\a> crniPW + \\cnNij) rn.n^Vll}, since 11*11^ Â£ Vh. In order to bound the first term in (2.55), we use (2.51) of Lemma 2.13 and (2.30) of Lemma 2.6 to get WW jcnNii>\\ < \\Â£ip n^'vil + \\nNCip Â£nNip\\ < ||g. -nw3s|| + r dip (2.56) jCi r C2 n HÂ£s9sII 7V2 r dip Cs dz To bound the second term in (2.55), observe that PHzv = IIzPv for any v Â£ V, since P operates only on fi. Therefore, denoting T -(I P) + aP, we have \\Â£U.Nip Â£n.zUNip\\ < C^SfiJlNip- ^ShUzILniP + \\TKNip THzUNip\\ d_ dz -S(iuNip nz-SfiiLNip S + \\TUniP n.THivV-ll. Applying (2.27) to bound the last two terms leads to \\Â£uNip jCnziiNip\\ Qm+l 2 dzm+1 e S/jJIpjip + Chn dm dzm TUniP < Chm Iljv dmip dzm CHn dmip dzm where we used the V-ellipticity of the bilinear form a(-, ) in the last step. We proceed further by using (2.51) of Lemma 2.13 and the fact that Iljv is an orthogonal projection ((i) of Lemma 2.6) to get \\Â£uNip Â£uziiniP\\ njvÂ£ dmip dzm + , dmip - n^Â£] dmqs dzm , C4 hm + VCIN* dm+1ip Ls dzm+' (2.57) 41 Finally, substituting (2.56) and (2.57) into (2.55) results in (2.54). Remark 2.15 (Interpretation of error bound) In the following, we interpret the error bound (2.54) more closely. For diffusive regimes (e < 1), the exact solution of the continuous problem has the diffusion expansion2 (see (1.13) in Section 1.2.2) = while the the right-hand side in this case is assumed to have the form 2 = 2o (z) + efiqi(z) + 0(e2). Therefore, it follows that JZs4> = 0(e) and Â£,sqs = 0(e2), since qs = Sq = Pq + e(I P)q. Taking this into account, for the error e/, in (2.54) we get eft = yB:(0(Â£,)+Â£0W)+A(ai' dmqs dzn + Caj^O(s) Thus, the error in the zeroth moment \\P(ip Vft)|| is bounded by 0(hm)+0(e) and the error in the higher moments ||(I P)(ip ^>ft)|| is bounded by 0(ehm) + 0(e2). In particular, for diffusive regimes, where e is very small, convergence of the discrete solution is also assured by the above bound for small N, which is a reasonable choice in this case, since the exact solution is nearly independent of fx. Moreover, the bound in Theorem 2.14 directly gives the optimal order of conver- gence for the spatial discretization without the use of Nitsches trick (Johnson [25, p. 97]). For example, for piecewise linear elements, which means r = 1, we can choose m = 2 and get an 0{h2) error bound in the L2-norm under the regularity assumptions on ip required in the theorem. On the other hand, if e is close to so that e > hm and s > 1/N, then an error bound can be obtained more easily, since Mlv < i + 4 dv dz + i+4 Ml 2 1,0 Therefore, for any v G VTl (Hm+1([zi, zr] x H2([ 1,1]), Ceals Lemma (1.24) and the bounds in (2.27) and (2.30) for the interpolation error lead directly to fC~ f 1\1/2 \H~M\v + IM-n*n^lli.o i \ 1/2 / /-Y 1 + ?) hr||^||i,o + ^ m1 dmip (2.58) dzm 1,0/ However, we point out that this bound will blow up in the diffusion limit e 0 for fixed N and h. 2The assumption that the exact solution has a diffusion expansion would have simplified the proof of Theorem 2.14; however, we preferred to present here the more general version. 42 For the case e > ^=, which is not covered by Theorem 2.14, the error bounds in Section 2.3.2 can be used instead. The second main class of finite-dimensional spaces considered here are formed by functions that are piecewise polynomials in space z as well as in angle fi. This choice corresponds to a Finite-Element discretization in both space and angle. Because Section 2.3.2 contains error bounds for this class of discrete spaces that are valid in non-diffusive regimes, we concentrate in the following only on error bounds for diffusive regimes. Therefore, we assume in the following theorem, which combines Ceas Lemma and (2.33), that the exact solution has a diffusion expansion, which simplifies the proof. Theorem 2.16 (Finite-Elements in space and angle) Suppose that 0 < a < 1 and 0 < e < . Let T/,bea partition of the computational domain D = [zi, zr] x [1,1] into rectangles T= \zj,Zj+1] x [/i,/i+i] of maximum diameter h. To be able to handle the boundary conditions properly, we assume in addition that (zj, 0) and (zr, 0) are nodes of the triangulation. Let V be given as in (2.38) and define W1 := | vh E C(D); vh\T = Â£ Cp,y z? f V T E Th l 0^/2,7 Vh(zi,ft) = o for /I > 0, Vh (zr /i) = 0 for \i< 0 Suppose ip E V C\Hr+1(D) is the solution of (2.20) and let iph G Vh be the solution of (2.20) restricted to Vh. Further, suppose that the diffusion expansion ip(z, p) = is valid. Then we have \C< \\ip- i>h\\v < \r?rChr (|Vlff'-+i(c) + Hr\h-+i(d)) , (2.59) with C independent of e, a, and h. Proof. From Ceas Lemma (1.24), we have U-M\v <^feM-nhiP\\v < Ipfl^-UhTP) + {I P)ti{ip -uhip) + _(/ PM Jihip) 1/2 + m n^voif (2.60) <4 -Pn^-{rp Hhip) e az + (/ p)ii-^(ip n h*p) + ~(i p)(v> n^) + ITO-n^)|| . 43 By (2.33) we now bound any of the above four terms separately and use the fact that lift is as an interpolation operator linear, so that = Ilft0 + ellhR- Since (z) in the diffusion expansion of if> is independent of angle //, we conclude that Ilft^z) is also independent of fi. Therefore, Pfi= 0 = so that ip/^-HftVO 1 d (2.61) < ||^ii Ilft^Rd^!^ where the last inequality follows from (2.33). Using (2.33) for the second term in (2.60) results in (/-P)/Z^(V<-IIftf) 5: IH HfcVllfri(D) < C hr IVI Hr+1(_D)- (2.62) Because and Ilft^ are independent of angle, we have (I P) = 0 = (I P)TLh- Therefore, the third term in (2.60) can be bounded by 1 (I-P)W- IlftVO -(i-p)e{R-nhR) <||^-nft^|| < Chr + 1\R\Hr+l(D) < Chr\^>R\Hr+l^D^ (2.63) since h < (zr z{) = 1. Similarly, a bound for the last term in (2.60) is given by \m n^VOII < U n^ll < Chr+1 < Chr WW^y (2.64) Inserting (2.61), (2.62), (2.63), and (2.64) into (2.60) results in (2.59), which proves the theorem. Remark 2.17 (V-ellipticity constant Ce) Both error bounds (2.54) an Ce- According to Theorem 2.11 and Corollary 2.12, Ce = 0.012, so 1/Ce = 83.3, which is fairly large. However, we would like to point out that we simplified the proof of Theorem 2.11 by considering only the worst case a = 0. Without setting a = 0, (2.46) would change to i(u,u) > + (1-6) ||u|| 2cf(l a)VV7 -2(1 d)(l a)y(T35)^(rZ7j Q dv 8z HI. HI (2.65) which clearly shows that the V'-ellipticity constant Ce increases with a. To judge the quanti- tative behavior, we computed Ce for certain values of a using (2.65). The results are plotted in Figure 2.1. Already for a = 0.3, 1/Ce drops down to 7.04. 44 Ce vs. alpha Figure 2.1: V-ellipticity constant Ce and its reciprocal as a function of the absorption parameter a. 45 CHAPTER 3 X-Y-Z GEOMETRY In this chapter, we generalize the scaling transformation and the error bounds for the Least-Squares Finite-Element discretization, from one-dimensional slab geometry to three dimensions. Since the main focus of this chapter is on diffusive transport problems, in the following we use the parameterized form (1.14) of the transport operator C. In addition, we assume that the total cross section Ut is constant in space ( parameter e is constant on the computational domain D := 1Z x S1, where 1Z C IR3 is a region with sufficiently smooth boundary, for example, of class C1,1 (Grisvard [22, p. 5]), and S1 denotes the unit sphere. Further, we suppose throughout this chapter that a < 1. Moreover, in the following we restrict our attention without loss of generality to problems with vacuum boundary conditions (so g(r,Q) = 0 in (1.4)). Problems with inhomogeneous boundary conditions can easily be transformed to problems with homogeneous boundary conditions and different right-hand sides (Oden and Carey [45, p. 27]). As in the one-dimensional case, a scaling transformation is applied to the transport operator prior to the Least-Squares discretization to ensure accuracy of the discrete solution in diffusive regions. In the three-dimensional case, the scaling transformation and its inverse are given by S := P + e(I P)] S'1 :=P + j(I-P). (3.1) They have the same form as in the one-dimensional case with the only difference that r = e and that the L2-orthogonal projection P onto the space of functions that are independent of direction vector Q is. now defined by Pil> := J ip(r,Q) dn. (3.2) s1 After applying the scaling transformation S from the left to the transport operator C in (1.14) and dividing by e, the transport equation becomes W := -SXV = YV> + -(/ P)i> + aPif> = qs, e e e (3.3) with qs := Sq. Throughout this chapter, we denote the standard inner product and the associated norm of L2(1Z x S1) by u v* dQ, dr; R s1 |]ti|| := y/(u, u) Vu,v G L2(TZ x S1), where v* is the complex conjugate of v. Further, for u,v E C(D), we define the following inner product (u,v)v II TZ S1 -50- -Vu -50 Vv + Â£ Â£ (I-P)u.(I-P)v + Pu Pv dQdr, its associated norm IMIv := y/(u>u)v = 1 2 1 -SQ Vu Â£ + -(I -P)u + IM2 1/2 (3.4) (3.5) and the space V := {v E C(D); u(r, 0) = 0 for r E dlZ, and fi n{r_) < 0}, (3-6) where the closure is with respect to the norm ||-||y, so that it is a Hilbert space. As mentioned in the introduction, the Least-Squares variational formulation of (3.3) is given by (1.19). Our first goal is to show that the bilinear form a(-,-), defined in (1.19) is continuous and V-elliptic with constants independent of parameter e and a. From these results will follow not only the well posedness of problem (1.19), as outlined in the introduction, but also the accuracy of the Least-Squares Finite-Element discretization applied to it in the diffusion limit. 3.1 Continuity and V-ellipticity As in Chapter 2, we conclude from the Cauchy-Schwarz inequality that the bilinear form a(-, ) is continuous, since |a(u, u)| = |(Cu, Cv)\ < ||Â£u|| ||Â£v||. We now use the discrete Holder inequality to get ||Â£u|| < < Vs SQ Vu Â£ + (I-P)u + HPul 1 2 l, Vu Â£ + -Â£(I-P)u + Il-Pul \ 1/2 I2 =vS! IV I since a < 1 by assumption. Thus, for any u, v E V, K,tOI < Ce Hiy |M|V, (3.7) with Cc = 3. For the more difficult part of the proof of the V-ellipticity of a(-, ), we proceed in the same way as in the one-dimensional case. We first scale (3.3) in addition from the right by S to get Â£S = vV+ {I P) + ocPi> = qs, 47 (3.8) where -0 := S 1'0. Define the new space V and associated norm by where V S-'V, ||| :=||Q-Yt>f + ||v| Q := -SQS = (1 e) (PQ + QP) + eQI. Shortly, we will prove that the bilinear form a(u, v) := J J CSu LSv dtldr V u, v Â£ V n s1 (3.9) is V-elliptic. The K-ellipticity of a(-, ) in (1.19) will then follow in the same way as in Corollary 2.12. Before we do this, we first establish the following lemmas, which are generalizations of Lemma 2.2, Lemma 2.4, Lemma 2.9, and Lemma 2.10. Lemma 3.1 For all u, v Â£ V, and e < 1, we have (i) (Pu,v) = (u,Pv); and {(I P)u,v) = {u,(I P)v); P2 = P; and (7 P)2 = (I P). Thus P and (I P) are orthogonal projections. (ii) (Pu, v) = (Pu, Pv); and ((7 P)u, v) = ((7 P)u, (7 P)v); (iii) ||t)|| < -5t e (iv) ||P||-e||(7-P)w|| < |H|; (v) (fi Vv, v) > 0; (vi) (Q Vv, u) > 0. Proof. The proofs of (i), (ii), (iii), (iv) follow analogously to that in Lemma 2.2 and Lemma 2.9. To prove (v), we apply the fundamental Greens formula (Ciarlet [16, p.34]) to get (Q Vu, v) : n s1 //S2' Vv v dQdr We therefore have J J v Q Vv dQdr + J J v2 Q n dQds. n s1 on s1 (Q Vv, v) = i J J v2 Q-n dÂ£lds. 8KS1 48 Splitting the boundary dll x S1 into the parts T+ := {(r, Q) G dll x S1; 0. n(r) >0} and r~ := {(r, fi) G dll x S1; Q-n(r) < 0}, the boundary integral becomes //2 97IS1 H n dÂ£lds vZQ. 21 dfids + v2Q n dQds ?;2Q n dllds > 0, since v(r,Q) = 0 for (r, Â£2) G F and v G V. Thus, altogether we obtain (Â£2 Vu, u) > 0. To prove (vi), we observe from (i) that S is self-adjoint with respect to ). Con- sequently, we have (Q Vv, v) \ 1 1 SQS Vv,v) = (Q VSv, Sv) = (Â£2 Vv, v) > 0, where the last inequality follows from (v). Lemma 3.2 (Poincare-Friedrichs Inequality) Suppose e < 1 and let 11 C M3 be a bounded domain. Then, for any v G V, we have ||v|| < diam(7Â£) [|Â£2-Vt7|| < diam(7Â£) ||Q-Vu||. (3.10) Proof. For ri}rk G 11 let [Li, Lk] := {n : r = 2Ii + (1 s)rk, for 0 < s < 1} denote the line segment between and rk. Let arbitrary r Ell and Â£2 G S'1 be given. We define *1 := min{f G IR [r, r -(- fÂ£2] C 11} ^2 := max {2 G IR : [r, r + tÂ£2] C 11} n := r + tiQ,] 7*2 := r + t2Q. Then it is easy to see that Â£2 -2l(2Li) < 0. Taking into account the boundary conditions for v G V, we therefore have that 9(7^, Â£2) = 0, hence, r r v(r, Â£2) = Jd/ds = j 0 Vv ds, where ds denotes the arc-length differential along the line {r + tU,t G 1R}. Therefore, we conclude from Holders inequality that L L2 / U Ke,2)I < / |Â£2 Vu| ds < J [Â£2 Vu| ds < diam^)1/2 I j |Â£2 Vu|2 ds ni 211 Vi \ 1/2 / 49 Applying Fubinis theorem, it follows that 2 J J kfcil)|2 dQdr Li n s1 < diam(77)2 J J |fi-Vu|2 dtldr. 71 S1 R. S1 From the relation v = Sv and (iii) of Lemma 3.1, we thus have -SQ.S Vv e = diam(77) ||Q Vv 988, ||u[| < diam(77) ||fi Vu|| = diam(77) [|fi VSu|| < diam(77) which proves the lemma. Lemma 3.3 Suppose 0 < Â£ < 1. Then, for any 6 Â£ [0,1], there exists 6 > 0 such that H(b, S) := \ | ^ (5 (1 + |)6) 2 + 45(1 <5) + (l 0 6 + 5 J < 0. where s := jtf1/2 e(l 6)1(,2J In particular, for 5 < 0.875, we can choose 6 = 0. Proof. The only difference to the proof of Lemma 2.10 is that now s = j^1/2 e(l <5)1 ^2j instead of s jV^2 eV3(l 6)1/,2J Therefore, when 6 > 0.875, we use the assump- tion e < 1 to get s > 1 2yjd(l d) =: (3. Everything else is analogous to the proof of Lemma 2.10. We are now in a position to state the central result of this section. Theorem 3.4 (17-ellipticity of (-, )) Let a(-, ) and ||-||~ be given as in (3.9) and (3.8). Suppose that 0 < a < 1, 0 < Â£ < 1 and that the diameter diam(77) of the domain1 77 is 1. Then there exists a constant Ce> 0 such that, for all v Â£ V, 12 a(v,v) Vv + aPv + (/ P)u (3.11) Ce (||Q Vti||2 + ||n||2) = Ce \\v\\2~, where Ce = 0.012, which is independent of e and a. Proof. In the proof of Theorem 2.11, we replace QÂ§% by Q Vv and for P and a(-, ) use the definitions of this chapter. Then the proof of Theorem 3.4 follows exactly els the proof of Theorem 2.11, except that the Poincare-Friedrichs inequality (3.10) of Lemma 3.2 and (iv) of Lemma 3.1 are now used to get ||Q-Vu||2 > [V6 eVI <5]2 ||u||2 1This can be established by a simple transformation of the space coordinates r. 50 Therefore, s in (2.47) is replaced by s = bound the functions Gi and G2. [Vtf e-s/1 6 and Lemma 3.3 is applied to From the V-ellipticity of the bilinear form a(-, ), the V-ellipticity of the bilinear form a(-, ) follows immediately as in Chapter 2. We summarize this result in the following corollary. Corollary 3.5 ( V-ellipticity of a(-, ) ) Let a(-, ) and ||-||F be given as defined in (1.19) and (3.5). Suppose that 0 and that diam(P) = 1. Then there exists a constant Ge > 0 such that, for all v G V, a(v,v) > Ce ||v||y , (3.12) where Ce = 0.012, which is independent of a and e. 3.2 Spherical Harmonics Since a truncated expansion into spherical harmonics (Pjv-approximation) is used throughout this chapter for the the disretization in angle, we introduce here the spherical harmonics and summarize important properties that are needed for the error bounds. Recall that the associated Legendre polynomials are defined for / > 0 and m = 0,..., l by (Margenau and Murphy [39, p. 106]) jm (3.13) where P;(/i) is the (unnormalized) Legendre polynomial of degree l. By the formula of Rodrigues (Arfken [3, p. .554]) for the Legendre Polynomials given by this definition becomes 1 j/+m = (3.14) Expression (3.14) can be used to extend the definition of P,m(fi) to negative integer values of m. It follows that P,m(/r) and Pl~Tn(fi) are related by pfmoo = (3-i5) The associated Legendre Polynomials satisfy the following recurrence relations (Ar- fken [3, p. 560]): pr() = [(Z + mjP^/x) + (Z m + ljPfoOO] , (3.16) y/T^fiPTO*) = 27^1 [^(^-^i-f1^)]. (3.17) y/l-fPTijl) = ^-[(Z + mKZ + m-ljP^Oi) (Z m + 1)(Z m + 2)PI^1(/i)] . (3.18) 51 These recurrence relations, although derived in (Arfken [3]) only for positive integers m, remain valid for negative values of m. This can be easily checked by substituting in the relation (3.15) into the left and right parts of the recurrence relations. Further, the associated Legendre polynomials satisfy the orthogonality relation -1 (3.19) Based on the associated Legendre Polynomials the spherical harmonics are de- fined by (Arfken [3, p. 571]) Y,m(6, p) := (-l)m Ci,m P,m(cos(fl)) eimv, (3.20) where . (2l+l)(l-m)\ \l----n X,---- (/ + m)! Here, 9 denotes the polar angle with respect to the z-axis, while p denotes the azimuthal angle about the z-axis is. The spherical harmonics form an orthonormal basis of L2(51): In particular, 2tr 7r ^: J J Y(9, p) YlYl'*(9, p) sin(0) d9 dp SU' 8mm>, o o which, by letting L! := (6, p) and dQ := sm(ff)dedv | can be written as J-Ylm(a)Y,r'*(tt)dn = 6w6mm., (3.21) where Y'*(Q) denotes the complex conjugate of Y (H) From (3.15) it follows directly that Y,-m(Q) = (-l)mY,m*({2). (3.22) By the definitions of 6 and p, we have = (f2x, Qy, fiz) = (cos(v?) sin(0), sin(^) sin(0), cos(0)) = (cos(^)a/1 p2, sin(y>)\/l - (3.23) with fi := cos(0). The spherical harmonics satisfy the following recurrence relations. Lemma 3.6 (Recurrence relations for spherical harmonics) For all l > 0 and we have, nxY,m = Pl^Y^t1 l,mC -Pl,-mYÂ£? + Qy Y,m = i ((-l)A,m^r + + A.-mFTr1 - QzY = J^mY! + TJ^mY] j, (3.24) 52 where &I,m 7l,m (/ + m + 2)(1 + m + 1) 4(2Z + l)(2/ + 3) Pl,m (l m)(l m 1) 4(2/ 1)(2Z + 1) (/ m)(l + m) (2/-l)(2/ + l) Vl,m (l + m + 1)(1 m + 1) (2/ + 3)(2/+ 1) Proof. To prove the first recurrence relation, we replace cos(^) by ei so that Using for the first term recurrence relation (3.17) and for the second term relation (3.18), after simple but tedious calculations we obtain the first recurrence relation. The second recurrence relation follows in a way similar to the first. For the third relation, recurrence relation (3.16) is used. Since the spherical harmonics form an orthonormal basis of i2(S'1), every v Â£ Hr(TV) x i2(51) has an expansion of the form oo l C (r,Q)=S E hm(r)Yrm, with ^,m(r)= / w(r, ft)Y,m*01) dSl. (3.25) /=0 m=l qi For any v Â£ Hr(TZ) x iT2 (S'1), we define jv-i ; r ^Nv(r,n):= 53 53 l>m(r)Yr(Q), with ^,m(r) = j v(r, Q)Y;m*(i2) dSl (3.26) /=0 m= l S1 as the truncated expansion of v into spherical harmonics. To bound the error of the truncated expansion, in the following lemma we use the fact that the spherical harmonics are the eigenfunctions of the Laplacian operator on the unit sphere, so AnV,m(fi) sin 9 39 (Sm6dd __L_____ + sin2 9 dip2 y,m(fl) (3.27) = -l(l + l)Y,m( for / > 0 and m = / + 1,..., 0,..., l. Lemma 3.7 (Truncated expansion into spherical harmonics) Let /3 be any multi-index and recall that D@v := gxfi1dJyfild2?3 Suppose that v(r,Q) Â£ H^(H) X if2(51) and, for JV > 2, let IIjv be defined as in (3.26). Then \\AnDev\\ for l > 0; -/ < m < l, (3.28) ||v njwll < ^||Anu||, (3.29) 53 with C independent of v and N. Proof. By the definition of 4>i,m and (3.27), we have 2 1 [/(/ + i)F (lDfv 1 [/(/+1)]2 2 where we applied integration by parts in the last step and took into account the fact that the boundary terms vanish. Since ||T(m||L2(5i) = 1) then the Holder inequality implies that ||^m||<^||A.vD'4- Further, since the spherical harmonics form an orthonormal basis, we have CO l CO l A ||w-hhI2 = E X / \ ]=Nm=-li l=Nm=-l Using (3.28) now results in lu i - ||v njvv|| <-||An|| X JX^ [/(/+!)] = X[/(/ + l)]2' CO f 21+1 Since < js and < jf for N > 2, then we can bound the sum in the following way: V 2/ + 1 f. 2 f 2 (^-1): < N2 so that ||v njvv|| < ||Anv|| 3.3 Error Bounds In this section, we use continuity and V-ellipticity of the bilinear form a(-, ) and Ceas Lemma to bound the error of a Least-Squares Finite-Element discretization applied to problem (1.19). The discretizations considered here are based on finite-dimensional spaces 54 with functions that have a truncated expansion into spherical harmonics with respect to direction angle Q and are piecewise polynomials of degree Iona triangulation of the region % into tetrahedrons. This choice of finite-dimensional spaces corresponds to a discretization by a spectral method in direction angle and a Finite-Element discretization in space. In transport theory, the spectral discretization in angle using spherical harmonics as basis functions is also called a Pjv-approximation. Let Th be a triangulation of 1Z into tetrahedrons T of maximum diameter h. For any u(r, Q) Â£ Hr+1(Tl) X L2{S1), let Il/jt; denote the interpolant of v by piecewise polynomials of degree r on the triangulation Th- Then, similar to (2.27), it can be shown (Ciarlet [16]) that, for 0 < m < 1, II nHL,0 < C hr+1~m Mr+1,0, (3.30) where ||-||m 0 denotes the standard norm of Hm(TZ) x L2 (S1) and | |r+i,o denotes the standard semi-norm of Hr+1(H) x L2(5'1) (see Section 1.4). In order to combine Ceas Lemma with (2.29) and (3.30) to obtain a discretization error bound, we need the following lemmas. Lemma 3.8 (Bound for commutator [!!//Â£ Â£II/v]) Suppose N >2 and let the operator An be defined as in (3.27). Let v Â£ H1(It) x H2(S1). Then there exists C > 0 independent of a and e such that ||[IIjvÂ£ Â£IIjv] v|| < |Anv|i,o. (3.31) Proof. By expansion (3.25), it easily follows that HpfPv = PUpjv and IljvPil Vu = Pfi VII/vu. Therefore, [HjvÂ£ Â£11*] = IM- V-fi- VBjv. (3.32) Now using the recurrence relation (3.24) in Lemma 3.6, we get OO / dv ox E E ^(/v^-zv^-r1 1=0 ml N 1 1=0 m=l N-2 1 f=0 m=l Similarly, we have 1=0 m=l 55 so that m=N N-l Ed^N 1 ,m / \rm1 ym+l\ dx (.-N-l.-mljv OCN-l,mYN ) . (3.33) m=JV+1 Similarly we get i(n',0'S_t* Â£}**) = m= JV+1 3j/ and N dv d nNnz -r qz ^-iijvv = Y dN,m vm 9N-l,r, m=N dz "Tat, Vm ST' vm (m7jy_i / , ^ ^AT-l,mJJV m=-JV+l Denoting by |77.| the L2-measure of 1Z it is clear that |]Y;m[| < \J\1Z\. Therefore, by (3.28) in Lemma 3.7, we can bound (3.33) as follows: dv 9 a dv a ox < Asit- ox 2y/W N(N + 1) N Y', @N,m + m= N 2 VW\ (N-1)N N-1 E m=JV+1 The sums can be bounded by f _ V l(N-m)(N-m-1) A (IV m) lV(2Ar + 1) 2-/ w-m 2^ \/ 4(2A^ 1)(2JV +1) ~ 9f'9Ar-1'\ 9f9/V-1'l - m=-N m=N 2(2AT 1) 2(27V 1) IV m=N and K yk1 l(N + m-l)(N + m) y^1 N + m (2JV 1)JV 4(2JV 1)(2JV H-1) ij+1 2(2* 1, 2(2JV-1) so that IljV fij; Clx "T oa: In the same way, it follows that di) d Djvfiy a--DyIl/yU dy dy < < *VW\ N dv An-r- ox N An dv dy 56 and dv d dv Hiv Clz- uz Unv dz dz < VW\ N N(N + TT X 7Nm + ' m=-N We now continue by bounding the sums in the following way: m* N M 0 N-1 ____ ^7Ar,m 2N 1 + 2N 1 ^ ^N2 -JV m=l s2j^i+2?n(w-1)w=w and so that V'1 - V'1 l(N + m)(N-m) ^ A . X, w i,rn X Y (2iV + 1)(2JV 1) ^ TJV> )VJ-1 m=-JV+l V ^ ^ A > m=N m=N+1 ----IlivU 0Z dz < 3^ ~ AT An <9u 3z which proves the lemma. Lemma 3.9 Let V and ||-||^ be given as defined in (3.8). Then for all v E V fl (H1( 1Z) x L2(51)): 1% < CIRIi.o. with C independent of e and a. Proof. By definition, it follows that 11% < ([(1 <0 {\m V|| + ||n -Pv.o11} + e\u- u||]2 + ||u||2) 1/2 Notice that lia-vfi|| = dv dv dv ^ dv dv dv < fbr-S- d x + + dz < v^3 ||u||1)0 since |fij;| <1, |f2y| < 1, and |fi0| < 1. Similarly we have ||Â£2 PVu|| < nxp dv dx + n P1 UyPdy + n*pl and < a/3 ||u||10 , ||Pfi-Vu||<||Q-Vu||||1)0. VN-l,m (3.34) 57 From these bounds, (3.34) follows immediately with C = ^[3%/3]2 + lj = \/28. Now we are in a position to establish the following error bound. Theorem 3.10 (Finite Element in space, Pm in angle ) Let 7^ be a triangulation of 1Z into tetrahedrons of maximal diameter h. Suppose 0 < a < 1, 0 < e < ^=, and diam(77) < 1. Let V be given as defined in (3.6) and let Vh be defined by Vh:=LhV: ft(r,Q) = f IPr(Th)\ , l 1=0 m=-l J where fPr(7ft) denotes the space of piecewise polynomials of degree < r on the triangulation 7ft. Let il> eVn (Hr+1(Tl) x H2(S1)) be the solution of (1.19) with qs e L2{11) x H^S1) and let tph 6 Vh be the solution of (1.19) restricted to Vh. Further assume that ip has the diffusion expansion ip(r,Q) = Q). Then U Mv (llA^s|| + |Antf|i,o) fc, +\ 7T C2 hr (|<Â£|r+i,o + |^ir|r+i,o), V '-'e with Ci and C2 independent of a and e. Proof. By Ceas Lemma, we have (3.35) H-M\v <\PÂ£ (\w n^v-llv + ||n^ nftnjvVll) V Og (3.36) The first term can be bounded by using the V-ellipticity of a(-, ), (3.29) of Lemma 3.7 and (3.31) of Lemma 3.8 in the following way: W-nNnv < -4= ll^(^-nJvV,)ll < Vcl 1 -4= (\\C1> - + IIPjvjC Â£11*] tf[|) (3.37) V^e < c c, We U l|An9i|1 + NMh To bound the second term in (3.36), we use that, by definition, ||i||v = \\t>\\y and that S~1EhUNip = TlhS-'ENip = Ilftl!*^ Therefore, \\ENip -EhEN^\\v = E-M-ip- lift II* V < C n*v> nftiLvV 1,0 58 where the last inequality follows from (3.34) in Lemma 3.9. We now use (3.30) and the diffusion expansion of ip to get nNip nhn^ < chr n^ = or V 1 Â£ n / 1,0 r+1,0/ since Plljv =: IIjvP. Remark 3.11 (Nondiffusive regimes) In Theorem 3.10, it is assumed that the analytical solution has a diffusion expansion in order to get an error bound in (3.38) with a constant that is independent of parameter e. For regimes where the diffusion expansion is not valid, j is of moderate size, so that there is no need for an error bound that is independent of e. Therefore, in this case, (3.38) can simply be bounded by so that the overall bound becomes lr+1,0 ' 59 CHAPTER 4 MULTIGRID SOLVER AND NUMERICAL RESULTS According to the theory derived in our earlier chapters, the Least-Squares approach yields accurate discrete solutions, even for diffusive regimes. In this chapter we confirm this very efficiently by a full multigrid solver. The following tests are restricted to the one-dimensional transport problem (1.6). For the discretization in angle, a Pjv-approximation is used, which is a spectral method using the first N Legendre polynomials as basis functions. For the discretization in space, slab (((j>f(z) G JPr(Th) with r = 1 or r = 2). Defining m := dim(IPr(7)l)) 1 and letting {rjk(z), k = 0,1,..., m} be a Finite-Element basis of IPr(T/t) (in the case r = 1 for example, r]k(z) are the usual hat functions), then the discrete space is given by for all / = 0,..., N 1 and all k = 0,..., m. For the computations in this chapter, we simplify the discrete problem (4.2) further by using a Gauss quadrature formula to approximate all integrations over angle fi, resulting in a Least-Squares discretization of either the 5V-flux equations or the moment equations. These two semi-discrete forms of the transport problem are introduced in the next section. result by numerical tests and demonstrate that the resulting discrete system can be solved we employ a Finite-Element discretization with linear or quadratic basis functions. To be more precise, we recall that the analytical solution has the moment expansion For the discretization, we truncate the sum N-1 h(z,fj.) = Â£>?(*) Mm) (4.2) 4.1 Sjy-Flux and P/v-i-Moment Equations 4.1.1 S'jv-Flux Equations Let Recall that, for a function v E VN we can use a Gauss quadrature formula with N support abscissas {fi\,..., and weights {wi,..., wjy} to write (4.3) since this quadrature with N support abscissas is exact for polynomials of degree < 2N 1 (Stoer and Bulirsch [49, p. 153]). In the 5jv-discretization, the flux is only computed for the discrete set of angles {fix,..., Hn}, so that the unknowns are given by the vector fp := / VfoMi) \ \ ^(z,hn) By collocating at the Gauss points and approximating the operator P by the sum in (4.3), the following SV-flux equations for ip can be derived from the transport equation (1.6): Hip M + CTt{lN ~ R) + VaR where q := ( 1 ^ , I := , := \ q(z,m) / l 1 ) =q, ( u l \ &N (4.4) M :=diag(/ii,...,//jv), i?:=WT. Further, for v E VN we note that the scaling transformation S = P + t(I P) in this notation becomes Sv = Snv := [R + t(Ijv R)] u, with In denoting the N x N identity matrix. Therefore, the scaled Sjv-flux equations are given by ILip := StfILip = 3 SnM + T(Tt(lN ~ R) + VaR 02 i> = q.> (4.5) with q := Sjv In., 0 V>(z() = 9i{v-i) \ 9i(v%) / and Q,Ii ip{zr) - / griUK+x) \ V 9r{m) J (4.6) respectively, where In denotes the ^ X y identity matrix. 61 4.1.2 Moment Equations In order to derive the moment equations from the flux equations, we note that, for V> G VN, N-l tP=^2Mz)pi(v)- (4-7) 1=0 The moments are given by 1 } N Mz) = o / (z>P)Pi{p)dP = VHz>N)Pi(Pi) (4-8) -'i where in the last step we again used the Gauss quadrature formula. Defining <Â£(z) := ((^0(z),..., ^jv_.!(2:))t and the matrices T and II as PDij := Pi-i(Pj), == diag(wi,...,uN), then the respective relationships (4.7) and (4.8) between the flux ij> and the moments can be written in matrix vector notation as l = TQ, and V>=TtÂ£. (4.9) In the following lemma, we summarize simple properties of the matrices R and T that we need for the derivation of the moment equations. Lemma 4.1 (Properties of R and T) We have: (i) wTl 1, so R2 = R and R(In R) = 0; (ii) Rt fl = QR; (iii) TQTJ = IN and TJTi2 = IN] (iv) Tfil = Tw = (1,0,..., 0)T and utTt = (1,0,..., 0); (v) Leting :*-* 3 the n r o bo 0 &0 0 h 0 TÂ£2MTt = 0 h 0 h 0 B NxN (vi) r i 0 .. 0 1 0 0 .. . 0 TQRTt _ 0 0 .. 0 NxN 62 Proof. N . 1 (i) : wTl = E = 2 / 1 dp = 1. Therefore, R2 = luTluT = l(wTl)wT = luT = R. i=i -i (ii) : RtQ ~ wl_TÂ£l = u>uiT = filwT = QR. N 1 (iii) : (TOTt) = Â£ Pi-i(Pk)ukPj-i{Pk) = \ f pi-i(p) pj-i(p) dp, = 6i:j. ,J *=i -l Therefore, TQTt = I, so Tr nonsingular => 3C such that TtC = I => TQTJC = TQ => C = TO => TtTO = I. N N 1 (iv) : (TO),. = J2 Pi-i(Vj)uj = E pi-i(Vj)ujPo(pj) = \ f pi-i(p)p0(p) dp = 6a. j=i i=i -i (v) : The unnormalized Legendre polynomials Pi(p) satisfy the recursion (Arfken [3, p. 540]) ppM = + ^^fl+iO*)- (4-10) Since the normalized Legendre polynomials are given by pi(p) := y/21 + 1 Pi(p), from (4.10) we have PPi(p) = bi-ipi^i(p) + bipl+i(p,) (4.11) with bi-i := \y-J2_1 Using (4.11), we then have (MTT)i,j = ViPj-ifai) = h3-iPi-z(Pi) + bj-iPj(Pi)- Therefore, N . [TQMTr]. = Y^Pi-i(Pk)uk (bj-2Pj-2(Pk) + bj-ipj(pk)) k=\ 1 1 = bj-Jpi-i{p)pj-2{p) dp + bj-v^ Jpi-i(p)pj(p) dp -1 -1 bj2^i j'1 T bj 15* j y -(_ 1. (vi): mRTr = (TO 1) (wtTt) = Tu utTt = (1,0,..., 0)T(1,0,..., 0), where we used (iv) in the last step. Multiplying the unsealed flux operator IL by TT from the right and by TO from 63 the left and using Lemma 4.1 gives TQILTt = TQMTt ^ + ( ' 1 0 . . 0 ' \ ' 1 0 . . 0 ' Bk+' In - 0 0 . . 0 + CTfl 0 0 . . 0 0 o , 0. ) _ 0 0 . 0. (4.12) ~Btz + =: JM, where 1M is the unsealed moment operator. Therefore, multiplying (4.5) by TO from the left and using (4.9) results in the following scaled moment equations: TOZLV> = TnsNTTmmTr = (TÂ£lRTr + tTOTt tTQRT7) / 1 T \ 0 + tIn - 0 V 0 0 ) JMj>_ (4.13) IMh =: TMj>_ = TO<^ Again using relation (4.9), it follows from (4.6) that the boundary conditions for the moment equations are given by la, 0 2 TT{Zl) = ( 9l(j*l) \ \ gi(fJ-K) / 0 ,Ia\TTl(zr) = / gr(x+1) V 9i(vn) (4.14) We conclude that the SV-flux equations and the P/v-moment equations are equiv- alent semi-discrete forms of the transport equation. The difference between these two sets of equations is that the non-derivative part in the flux equations is fully coupled, while the derivative part is decoupled. For the moment equations, the reverse is true. 4.1.3 Least-Squares Discretization of the Flux and Moment Equa- tions After deriving the Â£V-flux and moment equations we return to the discrete P/v- problem (4.2). Using a Gauss quadrature formula with weights {wi,...,wjv} and points 64 {fii,..., fix} to approximate the integration over angle n results in Zv N / pi)] (fij) [Â£bki,(z, fi)] (Hj)dz 1 i=1 ZI N I i=i (4.15) We note that on the left-hand side, the approximation of the integration over angle jx by the Gauss quadrature formula is exact as long as / < N 1: since then Cbk,i is a polynomial in fj, of degree l + 1, while Lijih is a polynomial in fi of degree N, then the product is a polynomial in fi of degree < 2N 1, for which the Gauss quadrature formula with N support abscissas is exact. Therefore, only in the equations for i, k = 0,..., m} must we introduce an error on the left-hand side by approximating the integration over angle. On the other hand, the same argument shows that the right-hand side is represented exactly by the Gauss quadrature formula as long as qs(z,fi) has an expansion into the first N 2 Legendre polynomials. With the notation introduced in Section 4.1.1 we have ( [Liph{z,ii)\ (m) V [Abh(z,n)]{nN) = and [Cbkti(ztfi)](fxi) > [Â£bki,(z,n)}(/iN) JLr}k(z)tJ+1, where tj+l denotes the (/+l)-nth column of the matrixTT defined in Section 4.1.2. Denoting by (,')jftN the standard Euclidean inner product of 1RN, (4.15) then becomes Z r j (MLh,ILT)k(z)ti+1)mN dz Z\ ZT = J dz Zl (4.16) for all k G {0,1,..., m} and / G {0,1,..., N 1}. Since the columns of TJ span 1RN, then we can substitute {t^,.. by the canonical basis {e^ ..., ew} of IRN and we recognize that (4.16) is a Least-Squares discretization of the SW-flux equations using the discrete space ' / Vh{z,fl i) > < N m jl Hft = >v-h = J2Y,vik,ik(z)z-j \ vh(z,fxN) y O II il II (4.17) This is the space of iV-vector functions whose components are piecewise linear (for r = 1) or piecewise quadratic (for r = 2) polynomials on the partition Tk of the slab. 65 Using (4.9) and (iii) of Lemma 4.1, we can rewrite (4.16) as J (pÂ£TTlh,TTTnÂ£TTTntf+1r)k(z)}iRN dz *1 Zr = J(^,TTmmTTmtJ+1r,k(z))MN dz Zi for all k G {0,..., m} and / G {0,..TV 1}, which by (4.13) and (iii) of Lemma 4.1 is equivalent to 2r j (Mlh,Mem{z))^ dz (4.18) Zr = J (TQ.qs,Meir)k{z))^ dz. Hi This is a Least-Squares discretization of the moment equations using the discrete space ' f &%(z) > m jl o 1* d) i-h . for l = 0,..., N 1 ^ N-l(z) J k=0 J All computations in the following sections are based either on discrete problem (4.16) or (4.18). 4.2 Properties of the Least-Squares Discretization In this section, we use the results of numerical experiments to observe properties of the Least-Squares discretization. The results plotted in Figure 4.1 and Figure 4.2 demonstrate the accuracy of the Least-Squares discretization in combination with the scaling transformation for diffusive regimes. The test problem we chose here is the same one used by Larsen et al. in [32]. The exact solution of the corresponding diffusion equation is (z) = 3/2z2 + 15z, which is plotted in solid in Figure 4.1 and Figure 4.2. The scalar flux 0 '= Piph of the solution iph of the Least-Squares discretization of the scaled transport equation using piecewise linear elements in space is shown by the crosses. For the problem in Figure 4.1, where the absorption cross section is zero, we used r = 1/of = e2 as the scaling parameter, which gives a higher accuracy than the scaling with r = e. An explanation of this result is given in the analysis presented in (Manteuffel and Ressel [38]). For the test problem in Figure 4.2, where cra ^ 0, the scaling parameter was chosen to be r = \/(Ta/crt = e^/a. Taking into account the fact that the mesh size of 1.25 is order of magnitudes larger than l/ both cases the results are very accurate. Further, we mention that the Least-Squares discretization without the scaling trans- formation results in the zero solution for both cases as indicated by the asterisks in Figure 4.1 and Figure 4.2. This outcome confirms the asymptotic analysis in Theorem 2.1, according to which the scalar flux of the Least-Squares discretization of the unsealed transport equation with piecewise linear basis elements in space is a straight line connecting the values at the boundary in diffusive regimes. Moreover, the asymptotic analysis in Theorem 2.1 asserts that the Least-Squares discretization of the unsealed transport equation using piecewise polynomials of degree > 2 in space has the correct diffusion limit. This too is supported by the observed maximum errors for a Least-Squares discretization of the unsealed transport equation with piecewise quadratic elements in space, which we list in Table 4.1. However, using the scaling transformation in combination with the Least-Squares discretization with quadratic elements in space achieves dramatically better accuracy in the discrete solution. For piecewise linear elements in space, the error bound in Theorem 2.14 indicates an 0(h2) behavior of the Least-Squares discretization error for a sufficient smooth solutions. To analyze the order of the Least-Squares discretization numerically, we used a problem with smooth exact solution sin(7rz). We then computed the discrete L2-error of the Least-Squares discretization with linear elements in space for a sequence of grids that were created from the coarsest grid by halving the mesh size from one to another grid. Table 4.2 depicts the ratio of these errors for each two consecutive grids. The value of approximately 4 of this quotient confirms numerically an 0(h2) behavior of the discretization error for linear elements. The solution of the transport equation is physically a density distribution and should therefore always be positive. The Least-Squares discretization has the drawback that it does not in general guarantee a positive solution. This is shown by the example in Figure 4.3, where the exact solution of the corresponding diffusion equation is again plotted as a solid line and the discrete Least-Squares solution is depicted by the crosses. Of course, this boundary layer can be resolved by refinement of the mesh, as shown in Figure 4.4. However, in the region [2,10], the solution is nearly constant, so that a refinement makes sense only in the region around the boundary layer. Therefore, the aim is to use adaptive refinement, which can be combined very naturally with a full multigrid solver (McCormick [42]). One easy criterion for determining the area of further refinement would then be to check where the solution is negative. Of course, this has to be combined with more sophisticated criteria that compare the solution of consecutive grids, for example. Besides having the correct diffusion limit, a discretization for transport problems must satisfy the extra condition to resolve, with a suitable fine spatial mesh, interior bound- ary layers between media with different material cross sections. To test numerically if the Least-Squares discretization meets these extra conditions, we used the test problem from (Larsen and Morel [33]), which is given in Figure 4.5. The solid solution plotted in Fig- ure 4.5 is computed by a Least-Squares discretization using 50 cells in both [0,1] and [1,11]. This solution approximates the exact solution plotted in (Larsen and Morel [33]) fairly well. We see further that the boundary layer is not resolved fully when the mesh spacing for the Least-Squares discretization is too coarse (crosses in Figure 4.5). In addition, the Least- Squares solution itself indicates an error by becoming negative. Again adaptive refinement would be an appropriate remedy. 67 Scalar Flux dib /i^- + 100(J-P)V> = 0.01 oz -0(0, //) = 0 for fi > 0 -0(10, //) = 0 for fj, < 0 (e = 0.01, a = 0.0) Solution of corresponding diffusion equation: cf>(x) = 3/2 x2 + 15s. Figure 4.1: Scalar Flux of exact (solid) and Least-Squares solution with scaling transfor- mation (crosses) and without (asteriks). 68 Scalar Flux (e = 0.01, a = 1.0) Solution of corresponding diffusion equation: = 3/2 x'2 + 15*. Figure 4.2: Scalar Flux of exact (solid) and Least-Squares solution with scaling transfor- mation (crosses) and without (asteriks). 69 Table 4.1: Comparison of maximum error for scaled and unsealed Least-Squares discretiza- tion with piecewise quadratic elements in space. OL 0.0 OL 1.0 l/h scaled unsealed scaled unsealed 4 1.8-10-4 5.2-10-2 1.2-10-4 3.2-102 8 1.2-10-5 1.2-10-2 7.7-10-6 7.7-10-3 16 ^3 1^ O 1 3 1 O i-H CO 4.8-10-7 1.9-103 32 4.8-10-8 1 o T( 00 3.0-10-8 4.6-10-4 64 3.0-10-9 1.8-10-4 1.8-109 1.1-10-4 128 1.8-1010 3.8-105 1.1-1010 2.2-10-5 256 1.3-10-11 6.3-10-6 7.9-10-12 3.8-106 Test problem: [ftjl + vtil-P) + (z,n) =q for (z, y) 6 [0,1] x [1,1] < -0(0, /z) = 0 for fi > 0 > , V^IjZ-O = 0 for fj, < 0 where = 1000.0, (ra = q := (in cos(irz) + Exact Solution: = sin(Trz), Number of Moments: N = 2. 70 Table 4.2: Order of Least-Squares discretization for linear elements in space. Â£ = 1.0 Â£ = 0.001 a = 0.0 a = 1.0 a = 0.0 ||e2fc||3 ll^lh o= 1.0 IlCah [|2 II gfc. II a IMI2 2h kid limits IMh C2h l5ll IKIh 4 8 16 32 64 128 256 512 1024 2048 4096 8192 8.5-10-2 2.1 -102 5:4-10-3 1.3- 10-3 3.4- 10-4 8.5- 10-5 2.1-10-5 5.3- 10"6 1.3- 10-6 3.3- 10-7 8.2-10-8 2,0-10- 4.0 3.9 4.1 3.8 4.0 4.0 3.9 4.0 3.9 4.0 4.1 2.610 6.7- 10' 1.7- 10' 4.2-10' 1.0-10 2.6-10' 6.6-10 1.6-10 4.1-10' 1.0-10 2.5- 10 6.5- 10 3.9 3.9 4.0 4.2 3.8 3.9 4.1 3.9 4.1 4.0 3.8 1.5-10-2 3.8-10-3 9.7- 10-4 2.4- 10-4 6.1-10-5 1.5- 105 3.8- 10-6 9.5- 10-7 2.4- 10-7 6;o-io-8 1.4- 10- 3.5- 10-9 3.9 3.9 4.0 3.9 4.0 3.9 4.0 3.9 4.0 4.2 4.0 1.2-10-2 2.9- 10-3 7.5- 10-4 1.8-10-4 4.7- 10-5 1.2-10-5 2.9- 10-6 7.3-10-7 1.8- 10-7 4.6- 10-8 l.l-lO-8 2.8-10-9 4.1 3.9 4.1 3.8 3.9 4.1 3.9 4.0 3.9 4.2 3.9 Test problem: [a*If + V>(0,//) q for (z,n) G [0,1] x [-1,1] 0 for fi > 0 0 for n < 0 where at = 1, cra = q := fiir cos(wz) + Exact Solution: ij>(z,n) = sin(7r;z), Number of Moments: N = 4. 71 Scalar Flux (e = 0.01,. a = 10.0) Solution of corresponding diffusion equation: = 1^e20v^0.+ (X + e2ojo_i) 6~VZ- Mesh size: h ^, Moments: N = 4. Figure 4.3: Example of violation of the positivity property by the Least-Squares discretiza- tion 72 Scalar Flux (e.= 0.01, a = 10.0) Solution of corresponding diffusion equation: Mesh size: h = , Moments: N = 4. Figure 4.4: Refinement resolves boundary layer. 73 Scalar Flux ( dib + a-tip Pcrsip = 0.0 = 1 for fi > 0 . ip{ll,/j,) = 0 for // < 0 , 2 0 < z < 1 , r\_/ 0 0 < z < 1 100 1 < z < 11 an Number of Moments: N = 16. The solid-line solution is computed by the Least-Squares discretization using 50 cells in both [0,1] and in [1,11]. Figure 4.5: Behavior of the Least-Squares discretization in the case of interior boundary layers. 74 4.3 Multigrid Solver In this section we describe the multigrid solvers, that were developed for solving the problems resulting from a Least-Squares discretization of Sjy-flux (4.16) and moment (4.18) equations with piecewise linear elements in space. We refer the reader who is not familiar with multigrid methods to (Briggs [6]) for an introduction and to (Hackbusch [24]) and (McCormick [41]) for more advanced topics. Essential for the efficiency of a multigrid solver is the proper choice of its compo- nents, mainly the intergrid transfer operators, coarse grid problems, and relaxation schemes. The choice of the first two components is naturally given by the Least-Squares variational formulation: the sequence of discrete spaces V\ C V2 C C Vj = Vh determines the coarse grid problems since they are just the restriction of the variational problem to these discrete subspaces; the prolongation operator, which is a mapping from a coarse grid to the next finer grid in the grid sequence, is formed directly by composing the isomorphisms between the discrete spaces and their corresponding coordinate spaces with the injection mapping between 14_i and 14 (Bramble [5]), (McCormick [43]); and the restriction operators, which are mappings from a finer grid to the next coarser grid, are just the adjoints of the prolon- gation operators. Therefore, the only multigrid components that need to be chosen here are the sequence of discrete spaces and the relaxation. No relaxation scheme is currently in use for transport problems that smooths the error in angle and in space simultaneously. Thus, instaed of devising a multigrid scheme that coarsens simultaneously in space and in angle, we consider first applying the multilevel-in- angle technique of (Morel and Manteuffel [44]), which is based on a shifted source relaxation scheme. After reducing the degrees of freedom in angle, a multigrid method in space is used to solve the remaining discrete problem. Thus, here we consider only the development of a multigrid solver in space. For the discrete subspaces, we Use the Finite-Element spaces with linear basis ele- ments on increasingly finer partitions (halving the cells) of the slab. 4.3.1 Sn Flux Equations The stencil that results from a Least-Squares discretization of the Sn flux equations (4.16) with these Finite-Element spaces is given in Appendix A and shows full coupling in angle. This suggests the use of a line relaxation in angle, which updates all angles for a given spatial point simultaneously. The matrix that must be inverted for each spatial point for this scheme is of the form (see Appendix A) := (aifi + a2QM + az&M2) + (ci MwuTM + c2ljujJ) . The first part is diagonal, and the second has the rank 2 factorization (ci MujjjT M + c2uxjJ) = Thus, A{ can be cheaply inverted by the Sherman-Morrison formula (Golub and Van Loan [20, p. 51]). Our computational tests showed essentially no differences in the error reduction and smoothing properties of this line relaxation scheme for various different orderings of the spa- tial points. To save computational, we thus use this line relaxation scheme in a red-black fashion, since then the residual after one relaxation sweep is zero at the black points and 75 not need not to be computed for the restriction to the next coarser grid. This scheme is also more amenable to advanced computer aechitecture efficiency. The convergence factors for this multigrid algorithm 1 listed in Table 4.3, are com- puted in the following way. A problem with zero source term and and whose exact solution equal is zero is used in combination with a randomly generated initial iterate. Then 30 multi- grid cycles are performed and the convergence rate is computed from the geometric average of the per-cycle reduction factors of the last 20 cycles. We thus reduce the influence of the initial iterate on convergence and observe what tends to be the worst-case factors. Here we study the (1, l)-V-cycle, which uses one relaxation before and one after coarse grid correc- tion. Observed factors for coefficient a. Factors for (2,l)-V-cycles are also included. Such factors are sufficient to get a solution with an error on the order of the discretization error by one full multigrid cycle, as demonstrated by the results in Table 4.4. The additional V cycle on the finest level 10, performed subsequent to the full multigrid cycle, is reducing the error only by a small amount. Thus, we can conclude, that the error after the full multigrid cycle is completed is already on the order of the discretization error. 4.3.2 Moment Equations The Stencil for the Least-Squares discretization of the moment equation (4.18) is given in Appendix B. In the interior of the computational domain, it is a 15-point stencil that connects the neighboring spatial points and the two higher and two lower moments. At the spatial boundary, however, the stencil couples all moments. Therefore, we use a line moment relaxation, that updates all moments simultaneously for a given spatial point all moments simultaneously. Since the efficiency of the smoothing again is observed to be independent from the relaxation ordering, as in the Sn flux case, we use a red-black ordering of the lines. The convergence factors for this multigrid algorithm, are listed in Table 4.5. For very large values of at, this multigrid solver is more stable with regard to roundoff errors than the multigrid solver for the Sn flux equations. Even for values of 106, we get (1, l)-V-cycle convergence factors of order 0.1. Again, these convergence factors are sufficient to get a solution with an error on the order of the discretization error by one single full multigrid cycle, as demonstrated in Table 4.6. 1This algorithm was implemented in C++ and a special array class was designed for this purpose. 76 Table 4.3: Multigrid convergence factors for solving the flux equations. (l,l)-V-cycle O'* a = 1.0 a = 0.5 a = 0.25 a = 0.1 a 0.0 Id)0- 0.088 0.085 0.087 0.118 0.169 101 0.082 0.083 0.083 0.110 0.136 102 0.052 0.052 0.053 0.106 0.130 103 0.088 0.091 0.088 0.105 0.130 104 0.091 0.091 0.091 0.105 0.130 105 0.092 0.092 0.092 0.105 0.130 106 0.090 0.092 0.092 0.102 0.133 (2,l)-V-cycle 10 0.053 0.050 0.053 0.105 0.155 101 0.047 0.047 0.047 0.082 0.104 102 0.019 0.024 0.024 0.077 0.097 103 0.020 0.021 0.021 0.076 0.096 104 0.020 0.022 0.022 0.076 0.096 105 0.020 0.011 0.023 0.076 0.096 10s 0.019 0.023 0.018 0.077 0.099 Test problem: \p-jh + V>(0,/z) = 0 for (z,fi) 6 [0,1] x [1,1] = 0 for n > 0 = 0 for fi < 0 where aa = Exact Solution: iJ>(z,(j,) = 0. Initial Iterate: randomly generated grid function. Mesh size: h = Number of Moments: N = 8. 77 Table 4.4: Full Multigrid (l,l)-V-Cycle convergence factors for solving the SW-flux equa- tions. Test problem: + ip(0,n) = q for Â£ [0,1] x [1,1] = 0 for fj, > 0 = 0 for (i < 0 where cra = q := /iircos(Trz) + aa sin(7rz), Exact Solution: ip(z, /j.) = sin(7rz), Number of Moments: N = 4. 78 Table 4.5: Multigrid convergence factors for solving the moment equations. (1,1 -V-cycle a 1.0 a = 0.5 a = 0.25 a = 0.1 o o II S To0- 0.052 0.086 0.083 0.118 0.169 101 0.091 0.092 0.091 0.117 0.136 102 0.056 0.056 0.071 0.106 0.131 103 0.092 0.093 0.092 0.105 0.127 104 0.095 0.094 0.094 0.106 0.129 105 0.095 0.094 0.093 0.107 0.130 106 0.095 0.092 0.092 0.107 0.130 107 0.095 0.092 0.092 0.107 0.130 10 0.095 0.092 0.092 0.107 0.130 109 0.095 0.094 0.092 0.107 0.130 1010 0.095 0.094 0.092 0.106 0.130 (2,1' -V-cycle 0-( a= 1.0 a = 0.5 or = 0.25 a = 0.1 a 0.0 0.074 0.051 0.054 0.105 0.155 101 0.055 0.055 0.055 0.082 0.104 102 0.025 0.025 0.039 0.077 0.097 10 0.023 0.026 0.042 0.076 0.096 104 0.023 0.023 0.042 0.076 0.096 105 0.023 0.023 0.042 0.076 0.096 106 0.023 0.023 0.042 0.076 0.095 107 0.023 0.023 0.042 0.076 0.095 10 0.023 0.023 0.042 0.076 0.095 109 0.023 0.023 0.042 0.076 0.095 1010 0.023 0.023 0.042 0.076 0.095 Test problem: [l*-jfc + V>(0,/r) = 0 for (z, fi) 6 [0,1] x [-1,1] = 0 for fi > 0 = 0 for n < 0 where cra = ^. Exact Solution: i>(z, fi) = 0. Initial Iterate: randomly generated grid function. Mesh size: h = Number of Moments: N = 8. 79 Table 4.6: Full Multigrid (l,l)-V-Cycle convergence factors for solving the moment equa- tions. Test problem: [Vlh + (z, fl) V(O^) 0(1. A*) = q for (z, fi) G [0,1] x [1,1] = 0 for fi > 0 = 0 for n < 0 where cra = q := fnrcos(irz) + cra sin(7r.z), Exact Solution: ip(z, fi) = sin(7rz), Number of Moments: N = 4. 80 CHAPTER 5 CONCLUSIONS 5.1 Summary of Results In this thesis, we have studied a systematic Least-Squares approach to the neutron transport equatio. The Least-Squares formulation converts the first-order transport problem into a self-adjoint variational form, which makes it accessible to the standard Finite-Element theory. Essential for this theory is the V-ellipticity and the continuity of the variational form, which leads directly to the existence and uniqueness of the analytic and discrete solutions and to bounds for the discretization error for a variety of different discrete spaces. Moreover, the variational formulation guides in a natural way the development of a multigrid solver for the resulting discrete problem. However, due to special properties of the transport equation, the Least-Squares approach is less straightforward than it first appears. In this thesis, we focused on neutron transport problems in diffusive regimes. In these regimes, the transport equation is singularly perturbed and its solution tends to a solution of a diffusion equation. Therefore, to guarantee an accurate discrete solution, a discretization for the transport operator is needed, that becomes a good approximation of the diffusion operator in diffusive regimes. Only a few conventional discretization schemes are known to have this property. By an asymptotic expansion, we show in Theorem 2.1 for slab geometry that a Least-Squares discretization with piecewise linear elements in space fails to be accurate in diffusive regimes. The choice of linear elements in space will for any right-hand side always result in a straight line connecting the prescribed values at the boundary for the principal part of the solution, which is independent of direction angle fi. Numerical tests confirm this behavior. On the other hand, we prove in Theorem 2.1 that, if piecewise polynomials of degree > 2 are used, then the principal part of the discrete Least-Squares solution becomes a Galerkin approximation to the correct diffusion equation in diffusive regimes. This means that the Least-Squares discretization will be accurate in this case. Numerical tests with piecewise quadratic elements again confirm this result. Because of Ceas Lemma, the Least-Squares discretization can be viewed as the best approximation to the exact solution in the discrete space with respect to the Least-Squares norm ||Â£-||, where C denotes the transport operator. In diffusive regimes, the different terms in the transport operator become totally unbalanced, which means that different parts of the solution are weighted much differently by the Least-Squares norm. With P denoting the L2-orthogonal projection onto the space of functions that are independent of direction angle, it is clear that the Least-Squares norm in diffusive regimes hardly measures the components Pip of the solution ip, although this is the main component for these regimes. The idea is therefore to scale the transport operator prior to the Least-Squares discretization, with the effect of changing the weighting in the Least-Squares norm. Clearly, the scaling from the left by S = P + r(I P) with r = 0(1/ solution component Pip. Numerical tests show that a Least-Squares discretization of the scaled transport equation, even for piecewise linear elements in space, yields an accurate solution in diffusive regimes. Moreover, they show for piecewise quadratic elements that the scaling transformation dramatically increases accuracy. The major part of this thesis is devoted to proving that the Least-Squares dis- cretization in combination with the scaling transformation S gives for a variety of simple Finite-Element spaces always accurate discrete solutions, even in diffusive regimes. As men- tioned above, essential for bounding the error is the V-ellipticity and the continuity of the Least-Squares form with respect to some norm. It is easy to show that the scaled Least- Squares form cannot be bounded from below by a standard Sobolev norm. Therefore, the first obvious choice in the one-dimensional case is the norm (ll^fe-ll2 + II II2)1/2- With re- spect to this norm, we prove V-ellipticity and continuity of the scaled Least-Squares bilinear form and derive error bounds for discrete spaces that use piecewise polynomials in space and piecewise polynomials or Legendre polynomials in angle as basis functions. However, since the V-ellipticity and the continuity constants for this norm depend on at and aa, these bounds blow up in diffusive regimes. To prove the V-ellipticity and continuity with constants independent of cr* and aa, we use a scaled norm. Based on the V-ellipticity and continuity with constants independent of at and aa, we obtain discretization error bounds for the same discrete spaces mentioned above, with constants independent of at and aa. Thus, these bounds stay valid also in diffusive regimes. This result is generalized to three-dimensional x-y-z geometry for discrete space that use piecewise polynomials as basis functions in space and spherical harmonics as basis functions in angle. We conclude that the Least-Squares approach in combination with the scaling trans- formation represents a general framework for finding discretizations for the transport equa- tion that are accurate in diffusive regimes. Further, it naturally guides naturally the develop- ment of an efficient multigrid solver for the resulting discrete system. This is demonstrated in this thesis for slab geometry and piecewise linear elements. The developed multigrid solver for this discrete problem has convergence factors on the order of 0.1, so that one full multigrid cycle of this algorithm computes a solution with an error on the order of the discretization error. 5.2 Recommendations for Future Work Our numerical results show that, when simple discrete spaces in space are used, refinement is needed in order to resolve boundary layers. Therefore, the aim for the future would be to combine the full multigrid solver with adaptive refinement. On the other hand, with the V-ellipticity and the continuity given, it seems fairly straightforward to establish error bounds for more complicated discrete spaces that can better resolve boundary layers, including those of exponential or hierarchical type. Furthermore, generalization of the scaling technique to anisotropic transport prob- lems suggests itself. 82 BIBLIOGRAPHY [1] R.A. Adams, Sobolev Spaces, Academic Press, 1975. [2] R.E. Alcouffe, E.W. Larsen, W.F. Miller and B.R. 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Carey, Finite Elements, Mathematical Aspects, Volume IV, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. [46] R.K. Osborn, S. Yip, The Foundations of Neutron Transport Theory, Gordon and Breach, Science Publishers, Inc., New York, 1966. [47] A.I. Pehlivanov, G.F. Carey and R.D. Lazarov, Least-Squares Mixed Finite Elements for Second-order Elliptic Problems, SIAM J. Numer. Anal., Vol. 31, No. 5, pp 1368-1377, October 1994. [48] G.C. Pomraning, Diffusive Limits for Linear Transport Equations, Nuclear Science and Engineering 112, pp. 239-255, 1992. [49] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, second edition, Texts in applied mathematics, Springer Verlag, New York, 1993. [50] S. Ukai, Solution of Multi-dimensional Neutron Transport Equations by Finite Element Methods, Journal of Nuclear Science and Technology, 9(6), pp. 366-373, 1972. [51] G.M.Wing, An Introduction to Transport Theory, John Wiley and Sons, Inc., New York, 1962. 86 APPENDIX A FLUX STENCIL In this section, we derive the stencil for the Least-Squares discretization of the Sn flux equations (4.16) for piecewise linear elements. We assume that the slab is partitioned into zi zq < zi << zm = zr and denote by h* := z*, Zk-i the cell width of cell k. We are then looking for a discrete solution in the form m N m h(*)=EE^ = Y.Mz)> fc=0 j = l Jfe=0 where := ..., ^fc,jv)T and with Vizi+1) 0 elsewhere Zh-z hk ~j 2r,*jW II l = (0, ,o)T. Plugging (A.l) into (4.16) gives then (A.1) Â£ fc-1 Zk-1 k-1Zk-l for all (i,j) e {(i, i) : 1 < Z < rn, 1 < i < ivj U j(0, j) : 1 < j < j} U |(m, j) : y < j < ivj . Since j has support only in cell i and cell i + 1, then (A.2) is equivalent to zi *i+1 J (aiL>1Lnr,ij)]Rrd* + J {QIL>IL!h,i+ij)Kd* (*i) (*2) Z% zi+l f (qq ,ILt] \ dz + [ (dq ,TLr\1 \ dz J \ ** -r,hJ/lRN J \ is A*+WjRW (A.3) Zi-1 (*3) (*4) In the following, we consider the terms (*1) to (*4) separately. To (*1): Applying the substitution z = z z,-_i, we have Zi j (^,%r..)^dz = Zi-1 dz where h hi and i := -r,J Z := h** with ipl := V'fc_1 and i/,r := }P_k Then it follows that 11% j = \ T ~ T)] Afe.j + ^ [r<7t(J lwT) + cralwT] ze;- 1L = ^ [lwT + r(J lwT)] M ^ ^ [r7t(J lwT) + o-alwT] - z) + rz 88 so that J(niLt,ILrLri.)]RNdz o h [lwT + t(I lwT)] M (ifrr - , i [luj + t(I lwT)] Me^J +y ^ jr [r +Y [i^T + r(J ^-T)] M (^r ^j) X ['r<7<(/ i^T) + ^lw7] e_j ^ +y |j<7t(J lwT) + <7alwT] ipj, ^ [rat(I lwT) + +y [Tr, ^ [r = l ( lM^T + ^{1 -hLX)]M (r - , e;)^ +\ (Q KMIwt + rVtM(J lwT)] (3^ + V-J ^ ^ (fi [^aIwT + rVt(I lwT)] M (jfr. 3^) ,Â£, +/, ^SJ KliST + rV?(7 laT)] (=t + ==) .S,)^ Consider all possible j and recall that h = h{ and that T are the corresponding values in cell i which we denote by cra i, crt i, respectively r,-, and that 3^ = ^Pi_l,'Pr = 3^. We then 89 get the following contribution from (*1): 1 \(t? 1) Mum7M t?Q.M2] fli [W 0-o.i) (MuuJ wwtM)] + J - Ti*i,d ^T] j 4-1 + [(i T?) MuujJM + rfQ,M2] [(Ti(TtiQM] +T t7"2*?,.'0 + (ah Tiali) f i- To (*2); Applying the substitution z = z Zi, we have 2i+l -2* fc+l J (QIL&>1L?hj)KNdz> where now (/i := hj+i) and v> h z hi := * Therefore, ILrij . = [lwT + r(7 lwT)] Mej + ^ [r<7t(7 lu/r) + cralwT] (h z) e_j K, = ^ [lwT + r(7 lwT)] M ij^j + ^ [r 90 so that J(tlJL^lL^^dz o h [lwT + t(I lwT)] M ^ ^ ^ [lwT + t(I lwT)] Me^ +y [r +y [JmT + T'U lwT)] M (r V>() i [r +y -i^-T) + o'alw1"] t_v ^ [r<7t(J- lwT) + o-alwT] +y [r = x (Q [m^T + t2m(/-^ M (t- -&) .s#)^ [ + ^ (fi KiwT + T2r - ,5#')^ +4 (si KlMT + *VU luT)] (t + &) , Consider all possible j and recall that h = h8-+1 and that (cra,at,T) are the corresponding values in cell i + 1, denoted by (ca,i+i, the following contribution from (*2): 91 (tt- K1 T?+i) MuultM + r?+1QM2] + ^ [('r/+l<7*,i+l O'a.i+i) (MywT + uwJM) 27f+1o-M+1ftM] + ^Yi W+lO'M+l^ + K,i+1 T?+1Or2<+1) WWT] | Â£. + (T- [ft+i -1)M<^TM 7i2+1fiM2] [(r2+1trtii+i o-a.i+i) (Mwut wwtM)] +^r1 N+i^+i0 + Ki+1 ^+1^+1) ^T] } Â£+! To (*3): We have where ^(z) = SjvÂ£(z) = IwTÂ£(z) + r(7 lwT)g(z), / \ (*) := : \ q(z,Vn) ) Therefore, Zi / (nt.-Oa.u)*, * Zt-l *i = j (filwTÂ£, [hjT + r(I lwT)] Me^)^ dz Zi-1 + j (filwTg, [Tat(I lwT) + Zi- 1 + J (nT(I-lwT)q,[lur +T(I-lwT)]MejT]rti)JRNdz Zi + J 1wt)Â£, [r 92 Full Text PAGE 1 LEAST-SQUARES FINITE-ELEMENT SOLUTION OF THE NEUTRON TRANSPORT EQUATION IN DIFFUSIVE REGIMES by KLAUS JURGEN RESSEL B. S. (Math), Universitat zu Koln, 1985 B. S. (Physics), Universitat zu Koln, 1986 M. S. (Math), Universitiit zu Koln, 1991 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 1994 PAGE 2 This thesis for the Doctor of Philosophy degree by Klaus J iirgen Ressel has been approved for the Graduate School by Thomas A. Manteuffel / I j_f 7 Date PAGE 3 Ressel, Klaus Jiirgen (Ph.D., Applied Mathematics) Least-Squares Finite-Element Solution of the Neutron Transport Equation in Diffusive Regimes Thesis directed by Professor Thomas A. Manteuffel ABSTRACT A systematic solution approach for the neutron transport equation is considered that is based on a Least-Squares variational formulation and includes theory for the existence and uniqueness of the analytical as well as for the discrete solution, bounds for the discretization error and guidance for the development of an efficient solver for the resulting discrete system. In particular, the solution of the transport equation for diffusive regimes is studied. In these regimes the transport equation is nearly singular and its solution becomes a solution of a diffusion equation. Therefore, to guarantee an accurate discrete solution, a discretization of the transport operator is needed that is at the same time a good approximation of a diffusion operator in diffusive regimes. Only few discretizations are known that have this property. Also, a Least-Squares discretization with piecewise linear elements in space fails to be accurate in diffusive regimes, which is shown by means of an asymptotic expansion. For this reason a scaling transformation is developed that is applied to the transport operator prior to the discretization in order to increase the weight for the important components of the solution in the Least-Squares functional. Not only for slab geometry but also for x-y-z geornetry it is proven that the resulting Least-Squares bilinear form is contin uous and V -elliptic with constants independent of the total cross section and the scattering cross section. For a variety of discrete spaces this leads to bounds for the discretization error that stay also valid in diffusive regimes. Thus, Least-Squares approach in cornbination with the scaling transformation represents a general framework for the construction of discretizations that are accurate in diffusive regimes. For the discretization with piecewise linear elements in space a multigrid solver in space was developed that gives V-cycle convergence rates in the order of 0.1 independent of the size of the total cross section, so that one full multigrid cycle of this algorithm computes a solution with an error in the order of the discretization error. This abstract accurately represents the COiltent/ publication. PAGE 4 Acknowledgements List of Notation . CHAPTER CONTENTS 1 INTRODUCTION AND PRELIMINARIES 1.1 Introduction and Outline 1.1.1 Opening Remarks .. 1.1.2 Outline 1.2 Neutron Transport Equation and Diffusion Limit 1.2.1 Neutron Transport Equation 1.2.2 Diffusion Limit 1.3 Previous Work on Numerical Solution 1.4 Least-Squares Approach 2 SLAB GEOMETRY .. 2.1 Problems with Direct Least-Squares Approach 2.2 Scaling Transformation ... 2.3 Error Bounds for Nondiffusive Regimes 2.3.1 Continuity and V-ellipticity 2.3.2 Error Bounds ...... 2.4 Continuity and V -ellipticity with respect to a scaled norm 2.5 Error Bounds for Diffusive Regimes 3 X-Y-Z GEOMETRY 3.1 Continuity and V -ellipticity 3.2 Spherical Harmonics 3.3 Error Bounds 4 MULTIGRID SOLVER AND NUMERICAL RESULTS 4.1 SN-Flux and PN-1-ZVloment Equations 4.1.1 SN-Flux Equations 4.1.2 Moment Equations .... 4.1.3 Least-Squares Discretization of the Flux and Moment Equations 4.2 Properties of the Least-Squares Discretization 4.3 Multigrid Solver 4.3.1 SN Flux Equations 4.3.2 Moment Equations 5 CONCLUSIONS 5.1 Summary of Results 5.2 Recommendations for Future Work BIBLIOGRAPHY APPENDIX A FLUX STENCIL B MOMENT STENCIL v Vl 1 1 1 2 2 6 7 9 13 14 17 20 20 27 30 38 46 47 51 54 60 60 60 62 64 66 75 75 76 81 81 82 83 87 98 PAGE 5 ACKNOWLEDGEMENTS I wish to thank, first and foremost, my advisor, Prof. Torn Manteufl'el, for his academic and financial support, which made this thesis possible. His ready availability to discuss problems in deep detail, along w.ith his mathematical insight and creativity resulted always in helpful answers and hints, which formed the foundation of this thesis. Secondly, I would like to express my appreciations to Prof. Steve McCormick, who has been a consultant to this work since its beginning and whose expertise in multilevel algorithrns has proved invaluable. I wish also to thank the remainder of my committee, Professors Jan Tvlandel, Jim Morel and Tom Russell. Jan Mandel and Tom Russell taught me in their claBses the mathematical theory of Finite-Elements. From the many discussions with Jim Morel during my stay of 9 months at Los Alamos National Laboratory I gained a lot of insight into transport problems. I am also grateful to the Center of Nonlinear Studies at Los Alan1os National Laboratory for the financial support and the use of their facilities during this tirne. Particular thanks goes to Debbie Wangerin, who guided me through the jungle of rules imposed by the Graduate School. Moreover, I wish to express my gratitude to Dr. Gerhard Starke, who proofread most parts of this thesis and rnade useful comments. Last, but not least, I would like to thank all members of the Center for Cornputational Mathematics of the University of Colorado at Denver and rny friends Marian Brezina, Dr. Max Lemke, Dr. Jim Otto, Radek Tezaur and Dr. Petr Vanek for their friendship and support. v PAGE 6 LIST OF NOTATION For the most part, the following notational conventions are used in this thesis. Particular usage and exceptions should be clear from the context. Scalars, Vectors and Sets :c, y, z standard space coordinates. zr, Zr left and right boundary of the slab. fJ polar angle with respect to z-axis. rp azimuthal. angle about z-axis. Jl = cos(O). crt total cross section. fTs scattering cross section. (fa = (J't-crs, absorption cross section. s := l/rrt, in parameterization for diffusion limit. a CJ a = ca, in parameterization for diffusion limit. r := s or cV"Ct + c2 scaling pararneter. J-lj Gauss Quadrature points. Wj Gauss Quadrature weights. N number of Legendre Polynomials used in truncated expansion. h mesh size. r. := (x,y,z). dr := dxdydz, incremental spatial element. !} := (n" rly, n,) = (cos( PAGE 7 Multi-index Notation (J lfJI := (fJr, (J,, (J3). := fJr + fJ2 + (J3. .EhEf31EJyf:!28;)i3. Operators I p II, s L\.n IL IM Identity. := f, dfJ (in 1 D), := J8 .an (in 3D). := P + T(IP), Scaling Transformation. = P P). unsealed Transport operator in 1 D or in 3 D. :=sf., scaled Transport operator. := !:SpS = (1c)(Pp + pP) + spl. := 1sns. -Projection onto space of functions1 that have truncated expansion into Legendre polynomials or spherical harmonics. Projection onto space of functions that are piecewise polynomials of degree r on a partition of the slab [ zz 1 Zr]. Projection onto space of functions1 that are piecewise polynomials of degree 1' on a given partition. := d: [ (1 p2 ) d:], Sturm-Liouville operator. Laplacian in polar coordinates on S1 SN Flux operator. Moment operator. Functions 1/!(z, p), 1/!(r.,fl.) Pr(P) PI (I') p,m(p) Y,m(fl.) PAGE 8 Function Spaces IP, (D) polynomials on the domain D with degree smaller or equal to r. IV,.(T,,) piecewise polynomials of degree:::; ron 7J,. C0(D) space of continuous functions on D. c=(D) space of infinitely differentiable functions on D. L2(D) space of square Lebesgue integrable functions on D. V Hilbert space. vh discrete subspace of v. V := s-1(V), where s-1 is the inverse of the scaling transformation. Bilinear Forms {Y., !2.) IfF' standard Euclidean inner product of mn' z,. 1 (u,u) := J J uv dl'dx, (in 1 D). Zl -1 := J J uv' dfJ dr, (in 3 D). := (Cu, Cv). a( u, v) ii( il, ii) := (CSu, CSv), double scaled form. Norms \\u\\ 1\u\\k,o := norm of Hk(Gl) X L2(G,), wbere G1 := [z,,zr] and G2 := [-1, 1] in lD and G1 := 1(. and G, :=51 in 3D. \u\k,O := C[lfl=k \\Dfu\\2 )112 semi-norm of Hk(G,) x L'(G2). \1\\v Norm associated with V. \\ \v Norm associated with V. 11\v\11 = + 1\v\1')112 Matriees "'-:=(w1,:WN)T 1 := (1, ... 1)T, N elements. R := 1' "'T fJ := diag(w1, ... WN ). M :=diag(I'1::1'N). Vlll PAGE 9 CHAPTER 1 INTRODUCTION AND PRELIMINARIES 1.1 Introduction and Outline 1.1.1 Opening Remarks The Least-Squares approach represents a systematic solution technique that includes theory for wellposedness of the continuous and discrete problem, error bounds for discretization error and guidance for the development of an efficient solver for the resulting discrete system. Furthermore1 the Least-Squares approach is a general methodology that can produce a variety of algorithms, depending on the choice of the Lea..-::t-Squares functional, the discrete space and the boundary treatment. For many partial differential equations (PDEs), a straightforward Least-Squares approach is ill-advised, since it requires more smoothness of the solution than a Galer kin ap proach and results in a squared condition number of the discrete operator. l:lowever, problems of first order or elliptic problems with lower order terms are converted by a Least-Squares formulation into a self-adjoint problem. Moreover) by introducing physically meaningful new variables and transfOrming the original proble1n into a system of equations of first order, the s1noothness requirements of the solution can be reduced and squaring the condition number of the discrete problem is avoided. This first-order system Least-Squares (FOSLS) technique has recently been successfully applied to the solution of general convection-diffusion problems (Cai et a!. [10], [11]) and the Stokes equation (Cai eta!. [12]). In combination with a mixed finite element discretization it can even be worthwhile to apply this technique to self-adjoint second-order elliptic problems in order to circumvent the Ladyzhenskaya-BabuSka-Brezzi consistency condition (Pehlivanov et a!. [47]). The subject of this thesis is the extension of the Least-Squares approach to the solution of the neutron transport equation. This equation mathematically describes the migration of neutrons through a host medium and their interaction (absorption or scattering) with the nuclei of the host medium after a collision. The fact that the neutron transport equation is a already first order integra-differential equation rnotivates a Least-Squares approach. However, due to special properties of the transport equation, this is less straightforward than it might appear. In diffusive regimes, where the probability for scattering is very high while that for absorption is very low, the transport equation is nearly singular and its solution is close to a solution of a diffusion equation. To guarantee an accurate discrete solution in these regimes, a discretization of the transport operator is required that becomes a good approxirnation of a diffusion operator in diffusive regimes. Up to the present, only a few special discretizations have this property. Therefore, diffusive transport problems are hard to solve. The principal part of this thesis is devoted to the extension of the Least-Squares approach to transport problems in diffusive regimes. 1.1.2 Outline This thesis is organized as follows: Chapter 1 continues with an introduction to the neutron transport equation in Section 1.2, where also properties of this equation, especially PAGE 10 for diffusive regimes (diffusion limit), are discussed. Section 1.3 provides an overview of previous work on numerical solution of neutron transport problerns. Finally, in Section 1.4 the Least-Squares discretization and its associated standard Finite-Element theory are briefly reviewed. Chapter 2 deals with the application of the Least-Squares approach to one-dirnensional (slab geometry) neutron transport problems. In Section 2.1, we analyze why a Least-Squares discretization directly applied to the transport equation is not accurate in diffusive regimes when combined with simple discrete spaces that use piecewise linear basis functions in space. To cure this problem, in Section 2.2 we intorduce a scaling transformation is introduced, which plays a key role in this thesis and is applied to the transport operator prior to the dis cretization. For the scaled Least-Squares functional, V -ellipticity and continuity are proved with respect to a sin1ple un-sealed norm in Section 2.3. Here we also obtain simple bounds for the discretization, although they are only valid for nondi:ffusive regimes. 'This is a result of the V-ellipticity and continuity constants dependence on the total cross section CJt and the absorption cross section CJ a, which are the coefficients in the transport equation that deter Inine how diffusive a region is (see Section 1.2). For diffusive regimes, O't is very large, which causes the simple error bounds in the unsealed norm to blow up. With respect to a scaled norm, the V -ellipticity and continuity of the Least-Squares bilinear form with constants in dependent of Ut and CJ a is proved in Section 2.4, which is the central part of this thesis. For a variety of discrete spaces, the V -ellipticity and continuity with constants independent of Ut and era are the foundation for the discretization error bounds in Section 2.5, which remain valid in diffusive regimes. The first class of discrete spaces consists of spaces with functions that can be expanded into the first N normalized Legendre polynomials in angle and are piecewise polynomials in space, whereas the second class of spaces is for.med by functions that are piecewise polynomials in space as well as in angle. In Chapter 3 we generalize the theory of Chapter 2 to three-dimensional problems in x-y-z geometry. In Section 3.1 we prove the V -ellipticity and continuity of the Least-Squares bilinear form with constants independent of O't and O'a Section 3.2 gives an introduction on spherical harmonics, which are used as basis functions for the discretization in angle (PN-approximation). Finally, in Section 3.3 we establish error bounds for the Least-Squares discretization using spherical harmonics as basis functions in angle and piecewise polynornials on a triangulation of the spatial domain as basis functions in space. Chapter 4 presents numerical results for slab geometry. In Section 4.1 we introduce the discrete ordinates SN flux and the PN moment equations, which are serni-discrete forms of the transport equation. All numerical results in this chapter are based on a Least-Squares discretization of these forms using piecewise linear or quadratic basis elements in space. In Section 4.2 we summarize some properties of the Least-Squares discretization. In Section 4.3 we describe the components of full multigrid solvers, that were developed for the SN flux as well as for the moment equations and implemented in C++ under use of a special designed array class. We also include convergence rates for the multigrid solvers in this section. In Chapter 5 we summarize our results and give some recommendations for future work. 1.2 Neutron Transport Equation and Diffusion Limit 1.2.1 Neutron Transport Equation Transport theory is the mathematical description of the migration of particles through a host mediurn. Particles will move through a host medium in con1plicated zigzag 2 PAGE 11 paths! due to repeated collisions and interactions with host particles. As a consequence of these collisions and interactions! the particles are transported through the host medimnj which explains the name transport theory. Transport processes can involve a variety of different types of particles such as neu trons, gas rrwlecules, ions, electrons, photons, or even cars, moving through various back ground media. All of these processes can be described by a single unifying theory, since they are all governed by the same type of equation, a Boltzmann transport equation (Duderstadt and Martin [17]). The origin of this theory is the kinetic theory of gases, in which case the transported particles and the host particles are equal to gas molecules. For this situation Boltzmann formulated his famous nonlinear equation in 1877 (Broda [9]). Transport theory currently plays a fundamental role in many areas of science and engineering. For instance, the diffusion of light through stellar atmospheres (radiative transfer) and the penetration of light through planetary atmospheres is fundamental for astrophysics. Moreover, radiation therapy, shielding of satellite electronics, and modeling of semiconductors require transport calculations. Further, the transport of vehicles along highways (traffic flow problems) and the random walk of students during registration can be analyzed by transport theory. The transport of neutrons through matter, which is considered in this thesis, is a major part of transport theory, because of its importance for the design of nuclear reactors and nuclear weapons. In the mathematical description of the transport of neutrons through Inatter three main interactions must be taken into account. After a neutron collides with a host nucleus, it can either get built into the nucleus (absorption), or it can be reflected, so that its travel direction and its energy have changed (scattering), or it can cause fission, that is, the nucleus breaks into two smaller parts and neutrons and energy are released. Since th.e collision of a neutron with a nucleus and its result are uncertain, the mathematical formulation of neutron transport is based on probability. Therefore! the quantity computed in neutron transport is the expected density _iV(r:j 0., E, t) of neutrons at position r: moving in direct;ion l1 with energy E at time t. Although the knowledge of the simple particle density P(L t), which specifies the density of neutrons independent. of their travel direction and their energy, would be sufficient for most applications! there is no equation that adequately describes this quantity. For this reason the density is further subdivided into the density N(.r:, E, t) of neutrons with a specific travel direction and a certain energy. Defining the phase space as the space spanned by the location vector and the velocity vector or equivalently spanned by the location vector, the unit vector describing the travel direction and the energy of a neutron, the density N can also be viewed as t.he phase space density. An equation for N(r:, E, t) may be obtained either in an abstract way from Liou ville's theorem (Cercignani [15]), (Osborn and Yip [46]), or from a simple balance argument, based on the neutron conservation within a small volume element of the phase space (Lewis and Miller [34]). Both ways result in the following balance equation for the expected neutron density N: &N -= -11 v'VN-rr,vN + s. iJt --(1.1) Here, v = [jji_ is the speed of a neutron with mass mN that has energy E. The probability ym:;; that a neutron will collide with a nucleus while traveling along a path of length dl is given by O't dl, where CTt(.I:, E) is the so called total cross section1 Since the neutron density is, in most applications, much smaller than the density of the host nuclei, neutron-neutron 1The reciprocal of the total cross section is called mean free pat.h (mfp), since this is exactly the length that a neutron can travel on the average before encountering a collision. 3 PAGE 12 collisions can be neglected, so that O't is assumed to be independent of the neutron density N. Otherwise, the balance equation (1.1) would be nonlinear. The first term on the rigbt-hand side of (1.1) represents loss of particles due to streaming, the second term represents loss due to collisions, and s ::::: s(r, ?., E\ t) represents both, implicit sources of neutrons due to inscattering and explicit sources. This explains why (1.1) is a balance equation. In the following only steady state problems are considered. Letting 1/;(r_, D., E, t) := vN(r_,D., E, t) be the angular flux of neutrons, the steady state form of (1.1) becomes (1.2) Further, fission as a possible interaction is excluded in the following, so that, in addition to an external source, s includes a term s8 (r, Q, E) that describes the neutrons that are scattered into the direction Q and energy E from some other direction Q' and some other energy E'. The scattering source term can be modeled as (Lewis and Miller [34, p.35]) 00 s, (r_, l., E) := J J o-,(r_, D.' __,D., E' __, E)1/;(r_, D.', E') di:l' dE', 0 S' where S1 denotes the unit sphere in ill3 and the scattering cross section O's is the probability that a neutron will be scattered from D.' -+D. and E'-+ B. In most ca..-,es 0'8 depends only on the angle between D.' and il, so O's::::: O's(r,il' il, E'-+ E). If O's is also independent of' Q, the scattering is said to be isotropic; otherwise, it is called anisotropic. In most applications, it is appropriate to discretize in energy to form what is known as the multi-group-equations (Duderstadt and Martin [17, p. 407]) (l,ewis and Miller [34, p. 61]). After dividing the energy range 2 into subintervals [Ek, E' +dE] and denoting by 1f} the Hux of neutrons with energy in this interval, this discretization results in a set of so called single-group-equations n. \I?J} + "'' E* can be neglected, the energy range can be assumed to be a bounded interval of the form [0, E*]. 4 PAGE 13 which is an equation for the single-group angular flux 1/J(r_, m, to be determined for all points r_ = (x, y, z) in a region n c IR3 and for all travel directions l.l_ on the unit sphere S1 The operator P is defined by P 0 for 11 < 0 P,P(z,p) := j 1/;(z,p') dp'. -1 l (1.6) (1. 7) The boundary conditions in (1.6) specify again the inflow of neutrons into the slab, since at the left end z, of the slab the inflow directions are given by I' > 0, while at the right end z,. the inflow directions are given by J1 < 0. 1 -I I' f Figure 1.1: Computational domain for slab geometry In neutron transport theory it is cornmon to introduce the absorption cross section a a:= O't-a8 Then the transport operator in three-dimensions becomes L:=l.l_ +!7t(I-P)+e7aP, (18) and for slab geometry we have a [.:=I' az + !7t(I-P) + e7aP. (1.9) 5 PAGE 14 1.2.2 Diffusion Limit Simple approximations to the transport operator 1 based either on ad hoc physical assumptions such as Fick's law 3 (Lamarsh [27, pp. 125-137]) or on a P1-approximation (Case and Zweiffel [14, Section 8.3]), which assumes that the flux is has an expansion into the first two Legendre Polynomials, result in a diffusion equation. This indicates that diffusion theory is related to transport theory. Indeed, transport theory transitions into diffusion theory in a certain asymptotic limit, called the diffusion limit. When O't oo and;;:1, equations (1.4) and (1.6) respectively becomes singular. By dividing (1.4) or (1.6) by r71 it easily follows that the limit equation is (IP),P = 0, which is fulfilled for all functions that are independent of direction angle 0. and fl, respectively. Moreover, when cr1-+ oo and 1 in a certain way, the limit solution of (1.4) and (1.6)1 respectively, will be a solution of a diffusion equation. This was discovered more than 20 years ago in the work by Larsen [28], Larsen and Keller [31] and Habet!er and Matkowsky [23]. To be more specific, we consider first the slab geometry case and assume that the slab has length 1, which can be established by a simple transformation (see Chapter 2). Following the recent published summary of Larsen [30], we introduce a small parameter c and scale the cross sections and the source in the following way: q(z,fl) 1 (Tt --7 -j where a: is assumed to be 0(1). Then the transport equation becomes [ i} l l /;(z,p) := f1 az P) + wP 1/J(z,fi) = Eq(z,fl). (1.10) (1.11) In combination with the special scaling (1.10), the diffusion limit is then defined by the limit c-+ 0. The physical meaning of scaling (1.10) is that the total cross section is large, so the syste.m is optically thick, whereas the absorption cross section and the external source arc srnalL We note that there are many scalings, which are capable to express this physical situation. However scaling (1.10) stands out since the diffusion equation d 1 d(z) ---+ O'a(z) = Pq(z,p) dz 30'1 dz (1.12) is invariant under it in the sense that, if the substitutions (1.10) are inserted into (1.12), then the resulting diffusion equation is independent of the scaling parameter c. Since equation (Lll) is of singular perturbation form, general boundary layers can be expected. Therefore, the solution is decomposed as 1/J(z,fl) = 1/JI(Z,fl) + 1/Jn(z,fl), where '!/JJ denotes the interior solution some mean free paths away from the boundary, and j;B denotes the solution near the boundaries. To determine 1.h, the following asymptotic expansion ansatz (Friedrichs [19]) 00 PAGE 15 is inserted into (1.11). By equating the coefficients of different powers of, it can be shown (Larsen [30]) that 1/;0 is independent of angle, so 1j;0(z, p) = 1j;0(z), and that 1j;0 (z) is a solution of the diffusion equation (1.12). To obtain the boundary layer part "IPB, an asymptotic expansion of the boundary layer is perforrned) which is then rnatched with the interior expansion. It follows (Habetler and Matkowsky [23]) that the boundary layer part ,p8 decays exponentially with the distance from the boundary and that its leading tern1 is also independent of the direction angle. Altogether, this results in the following diffusion expansion 1/;(z, p) = q)(z) + s PAGE 16 Because of the property that the analytical solution of the transport equation IS converging in the diffusion limit to a solution of a diffusion equation in the interior of the slab, the discretization of the spatial dependence is much more difficult. For accuracy reasons, the discrete solution must have the same property. Therefore, this requires a discretization of the transport operator that becomes a good approximation of a diffusion operator in diffusive reg1mes. By applying the asymptotic expansion technique introduced in Section 1.2, to the discrete solution, Larsen, Morel, and Miller [32) analyze the behavior of various special discretizations in the diffusion limit. In the discrete case, the mesh size h has to be considered as a second parameter besides the parameter c. Therefore, they define in their work the following two different limits. If, for a fixed mesh size h, the discretization approximates a diffusion operator in the lirnit c -r 0, then the discretization is said to have the correct thick diffusion limit. On the other hand, if the mesh size h varies linearly with c in the limit c -r 0 and this limit results in a consistent discretization of a diffusion operator, then the discretization is said to have the correct intermediate diffusion limit. Since standard finite difference discretizations, such as upwind differencing, fail to have a correct thick diffusion limit, special discretizations have been developed that behave correctly in diffusive regimes. Among them are the Diamond difference scheme and the difference schemes of LundWilson and of Castor) which, according to the analysis of Larsen, Morel, and Miller [32], give the correct thiek diffusion limit for the cell average flux. Moreover) Finite-Element disc.retizations have been applied for spatial discretiza tion of the neutron transport equation in different ways. The direct Galerkin approach to the first order integro-differential form (1.6) of the transport equation, as considered first theoretically by Ukai [50] and numerieally by Martin [40], does not have the correct behav ior in the difl'usion limit, except when special discontinuous finite elements are used. 'l'he use of discontinuous finite elements results, for exarnple, in the Linear Discontinuous (LD) discretization (Alcouffe et al. [2]) and the Modified Linear Discontinuous (MLD) scheme (Larsen and Morel [33]). MLD has the additional property that with a suitable fine spatial mesh, it can resolve boundary layers at exterior boundaries or at interior boundaries between media with different material cross sections. Further, a variety of Ritz variational formulations have been proposed (see Kaplan and Davis [26] for a summary). CI'hey have the self-adjoint second-order even-parity4 form of the transport equation as its Euler equation and lead, independent of the choice of the discrete Finite-Element space, to correct diffusion limit discretizations. However, the evenparity form of the transport equation is only valid for nonvacuurn regions and becomes very tedious for anisotropic scattering or anisotropic sources (Lewis and Miller [34, p. 260]). For the solution of the discrete system, a simple splitting iteration known as source iteration or transport sweep has been used in the past. Because of the slow convergence of this iteration for diffusive regimes (convergence 10( -$$),the Diffusion Synthetic a, Acceleration (DSA) method (Larsen [29]) was developed, which uses a diffusion approxima tion to accelerate the source iteration. By spectral analysis, Faber and Manteuffel [18] have shown why this method is successful for problems with isotropic scattering. However, for problems with anisotropic scattering, DSA is less effective. Tvloreover, rnultigrid methods have been employed for the solution of discrete neutron transport problerns. For the LD scheme, a multigrid algorithm in space was developed by Barnett, Morel and Harris [4] which proved to be effective even for highly anisotropic 4It can be shown [35] by a certain transformation that the even-parity form is closely related to the Least-Squares formulation considered here. 8 PAGE 17 problems. For isotropic problems, this algorithm is competitive with DSA, although it uses an expensive block-smoothing. The multigrid algorithm in space of Manteuffel, McCorrnick, Morel, Oliveira and Yang [36] for isotropic problems, discretized in space by the MLD scheme, employs a special operator induced interpolation and has been ported very efficiently to a parallel architecture [37]. For anisotropic problems a technique called multigrid-in-angle was developed by Morel and Manteufl'el [44]. This scheme involves a shifted transport sweep to attenuate the error in the upper half of the moments, so that the remaining error can be approximated by the solution of a problem discretized in angle based on only the lower half of the rnoments. Recursive application of this procedure leads to an isotropic problem on the coarsest level, which can be solved by a multigrid method in space. For higher dimensional problems, the discretization of the angle dependence also becomes a problem. For problems with isolated sources in a strongly absorbing medium, anomalies in the flux distribution, called ray effects (Lewis and Miller [34, p. 194]), are likely to arise in combination with a discrete ordinates (SN) discretization. The SN discretization causes a loss of rotational in variance, since this discretization transforms the fully rotational invariant transport equation into a set of coupled equations that are at most invariant under few discrete rotations. Thus: an azimuthally uniform flux, for example, is approximated by a set of 8-functions at discrete angles, which can be very poor if the number of discrete angles is not sufficiently large. One potential remedy is a PN discretization) which is a spectral Galerkin method using spherical harmonics as basis functions. This discretization results in a fully rotational invariant discrete problem. However) the coupling of the discrete equations is complicated and the treatment of boundary conditions is less straight forward. As in the one-dimensional case) for higher dimensions the discretization in space must have the correct behavior in the diffusion limit in order to obtain accurate discrete solutions for diffusive regimes. The direct extension of the appropriate one-dimensional disc.retizations is complicated, however. BOgers, Larsen and Ada1ns [7] have shown that the linear discontinuous (LD) finite element discretization on rectangles does not yield a correct diffusion limit discretization, whereas the MLD discretization does. However, the efficient solution of the discrete system resulting from the MLD discretization is an open problem. Applying a similar multigrid algorithm, which was developed by Manteuffel et a!. [36] for the one-dimensional case, would require the extension of the operator induced interpolation to higher dimensions, which is complicated. Morel et al. are in the process of developing a method for three space dimensions based on the even-parity form of the transport equation and using a PN discretization in angle. We conclude that an arsenal of highly specialized computational methods exists) whose design is adapted for particular transport problems. However, there is lack of a general systematic solution approach that includes existence theory of the analytic and discrete solution, error bounds for the error of discretization and guidance for the development of an efficient solver of the resulting discrete problem. Especially for higher dimensional problems, such an approach seems to be needed. 1.4 Least-Squares Approach In this section) we introduce a systematic solution approach to the neutron transport equation that relies on a Least-Squares self-adjoint variatio.nal formulation of (1.4), and we summarize the associated standard FiniteElement theory. The Least-Squares approach can be considered as a systematic solution approach: since it includes theory for the existence and uniqueness of the analytical and discrete problem: as well as bounds of the discretization error for a whole class of Finite-Element spaces. Furthermore, this approach will guide the 9 PAGE 18 development of a Multilevel Projection Method (McCormick [43]) for the efficient solution of the resulting discrete system. A Least-Squares Finite-Element discretization with piecewise linear basis functions in space directly applied to (1.4) does not have the correct behavior in the diffusion limit (see Section 2.1). For this reason, the scaling-transformation [P + r(l-P)], with parameter r E IR+ specified later, is applied to the transport operator prior to the discretization: C [P + r(I-P)J[l.J. \7 + o-,1-o-,P] = P(l.J. \7) + r(IP)(l.J. \7) + ro-1(1-P) + (o-to-,)P P(l.J. \7) + r(IP)(l.J. \7) + ro-,(1-P) +o-aF, (1.17) where in the last equation ( O'"t-cr s) was substituted by the absorption scattering cross section era. In this transforrned operator, the Least-Squares variational formulation of (1.4) is given by min F(w), EV with F(,P) := J J [/;(:c,l.J.)-q,(z:,l.J.)]2 drldr, (118) n S' with q, =Sq. The Hilbert space V with underlying norm II llv will be specified later. A necessary condition for 1/; E V to be a minimizer of the functional F in (1.18) is that the first variation (Gateaux derivative) ofF vanishes at 1j; for all admissible v E 'V, resulting in the problem: find lj; E V such that a(,P,v):= j jcwCvdrldr = j jq,CvdOdr VvEV. (1.19) n s1 n s1 The essential part of the theory is to show that the symmetric bilinear form a(,) in (1.19) is V-elliptic, i.e., there exists a constant Ce > 0 such that, for all v E 'V, a(v,v)?: C, (120) and continuous, i.e., there exists a constant Cc > 0 such that, for every u, v E V, la(u,v)l :": C, llullv llvllv(121) The proof of the continuity is straightforward, but the proof of the V-ellipticity is difficult and tricky. Denote the standard inner product and associated norm of L2('R x 5'1 ) by (u, v) := ./ J u v' dOdr; n S' I \fu, v E L2(1?. x 51), where v' is the complex conjugate of v. Using (1.20), (1.21) and the assumption that q,(z:,l.J.) E L2(R x 51), which ensures that the functional l: V ([); l(v) := J J q, Cv drldr n s' is bounded (!l(v)l :": c;1'llq,llllvllv ), then the Lax-Milgram Lemma (Ciarlet and Lions [16, p. 29]) can be applied. It follows that problem (1.19) is well posed in the sense that its solution exists, is unique and depends continuously on the data q8 The latter follows from 10 PAGE 19 so For the Least-Squares Finite-Element discretization of (1.19), the Hilbert space V is replaced by a finite-dimensional subspace V" c V, and (1.19) becomes: find >!Jh E Vh such that (1.22) The existence and uniqueness of a solution 1/Jh E Vh of the discrete problem (1.22) follows again from the Lemma of Lax-Milgram since, as a finite-dimensional space, Vh is a closed subspace of the Hilbert space V and is, therefore, also a Hilbert space with respect to the inner product of v restricted to v". By subtracting ( 1.22) from ( 1.19), it follows immediately that the error is orthogonal to vh with respect to the bilinear form a(-, l (1.23) The Cauchy-Schwarz inequality and (1.23) lead directly to Cea's Lemma (Brenner and Scott [8, p.62]): a( 1/J ,Ph, 1/J 1/Jh) :<; a( 1/J -vh, ,P -vh) I lib->!Jhllv :<; (C;c' min 111/J-vhllv, v c: vhEVh (1.24) with the use of the V-ellipticity (1.20) and the continuity (1.21). By (1.24), the problem of finding an estimate oftbe error is therefore reduced to estimating min II-1/JhllvThese vhEVh kinds of estimates are pl'ovided by approximation theory for a wide class of spaces Vh. For example, when we consider for simplicity only a semi-discretization in space where Vh is formed by piecewise polynomials of degree r, v = Hm(n) X 2(S1 ), vh c v, and the exact solution is in JfC+1(n) x L2(S1 ), it can be proved (Ciarlet and Lions [16, Theorem 16.2]) that (1.25) where h is the maxirnurn mesh size of the triangulation of n used and llvllm,O := [ L j j ID"vl'dlldr] 112 lo:l::;m R 51 denote the standard Sobolev norm and semi-norm (Adams [1]), respectively. Here, we use the standard multi-index notation 3 IPI = LiJ; i::::::l for ;3 := (;J1, ;3,, ;Js). 11 PAGE 20 For Vh forn1ed by piecewise polynomials of degree 1"1 the combination of (1.24) and (1.25) results in the overall error bound (1.26) The crucial point here is that we have shown V-ellipticity (1.20) and continuity (1.21) with respect to a weighted norm with constants Ce and Cc independent of crt and Ua1 so that an error bound similar to (1.26) for a discretization in space and in angle holds independent of the size of crt and cr a I-Ience1 the Least-Squares Finite-Element discretization of the scaled transport operator with piecewise polynomials of degree r 1 as basis functions yields an accurate discrete solution even in the diffusion limit. 12 PAGE 21 CHAPTER 2 SLAB GEOMETRY The Least-Squares approach is applied in this chapter to the one dimensional (slab geometry) neutron transport equation (1.6). Throughout this chapter, we assume without loss of generality the following: 1) The total scattering cross section is constant in space, so O"t(z) :.:::: CTt. This can be established by the transformation fO't(s)ds z' = "----. (2.1) O't(s) ds ,, The transport equation then becomes [11 + O';(I-P) + P] 1/;(z', 11) = q' (z', 11), with z, z, ' 0'; = J O't(s) ds, = ::i:i J O't(s)ds, '(' ) q(z,/1)1 ()d q z ,/1 = O't(z) O't s s. ,, Zt ,, 2) The slab has length 1, so (z,z,) = 1. If the transformation (2.1) was already applied, this is directly fulfilled; otherwise, this can be established with the simple transformation z" = -( ) This changes O"t, 17a, and q to Zr-Zl 3) We impose homogeneous (vacuum) boundary conditions, so m(l') =: 0 and g,(l') =: 0 in (1.6). This can be done in the following way. Define { g1(11) for 11?:: 0 1/!&(Z,/1) := g,(!') for 11 < 0 Then, clearly, 1/J,(z,/1) E H1([z1,z,]) x L2([-1, 1]), so that L1j;, is well defined and we can solve the problem l1fo = q -l?j;b with homogeneous boundary conditions. The original solution is then given by PAGE 22 2.1 Problems with Direct Least-Squares Approach In the following, we give an explanation as to why a Least-Squares Finite-Element discretization applied to (1.6) using piecewise linear basis functions in space does not, in general, yield a correct diffusion limit discretization. We recall that the Least-Squares vari ational formulation of (1.6) is given by and mm F(,P); with '" 1 F(1/J) := j j [l1/J(z,p) -Eq(z,p)]' dpdx, (2.2) 2! -1 V := { v(z, p) E L2(D): E I}(D), v(zl,!J) = 0 for I'> 0, v(zr, p) = 0 for I'< 0}. In (2.2) we used the parameterized form (Lll) of the transport equation, since it is better suited for a diffusion limit analysis. For the discretization of (2.2), the rninimization of the Least-Squares functional is restricted to a finite-dimensional subspace Vh c V. Without loss of generality, in the follow ing analysis for the discretization in angle we use a P1 approxi1nation) which assumes that the angle dependence of the solution has an expansion into the first two Legendre Polyno mials. One reason for this is that a semi discretization only in angle by a P1 approximation results in a diffusion equation [14) Section 8.3). Second) the behavior of the discretization in diffusive regimes, where according to (1.3) the exact solution is nearly independent of angle, is analyzed here; thus, a P1 approximation allowing a linear dependence in angle is sufficient. For the discretization in space, we use piecewise polynomials on a partition Th of the slab. Altogether, this results in the discrete space Vh := {vh EC0(D) :vh(z,p)=o(z)+tJ(x), where o,1 ElP,('TJ,); vh(z,,p) = 0 for I'> 0, vh(z,p) = 0 for I'< 0}, (2.3) where IPr(Th) denotes the space of piecewise polynomials of degree ::; ron the partition Tj1 of the slab. By the asymptotic expansion introduced in Section 1.2, the minimizer of the Least Squares functional can be characterized as follows. Theorem 2.1 (Characterization of Least-Squares minhnizer) Let the Least-Squares functional P and the discrete space Vh be given as defined in (2.2) and (2.3) respectively. Suppose V'h E vh minimizes F restricted to vh. Suppose further that E :S 1 and that 1/Jh has the asymptotic expansion in E given by 1/Jh(z, p) = + tJHz), with 00 =I; E"ryv(z); Hz):= I;c"bv(z), /)=::0 v=O where Tj11, 611 E IPr(Th) are independent of parameter c for all v. We then have: (i) b0(z) =e 0. (ii) %(z) = -b1(z). 14 (2.4) PAGE 23 (iii) Let U" := {rJo E IP,(Th): 7/o(z,) = 7/o(z,) = O,ryo fulfills (ii) for some 51 E .IP,(Th)}. Then for all 7/0 E U h: 1,,.1_,' d j'' d :J'lo'lo + aryoryo z = qryo z. zr zr Proof. We first prove (i). Using expansion (2.4) in (1.11) we have and, therefore, F(V;h) = L F,(,Ph), v::::-2 with (2.5) I, J j %(z)bo(z) + bo(z)b,(z) dz, ,, and F,(V;h) independent of for v 0. For :S 1, it is possible to bound F(,Ph) from above independent of E by (2.6) since lPh minimizes F and 0 E Vh Therefore, we must have F_z(lPh) = 0 and F_1(,P) = 0, since otherwise F( 1/Jh) ___,. oo in the limits -r 0, which contradicts (2.6). In combination with (2.5), we conclude that So(z) = 0. To prove (ii), by virtue of (i) we can restrict the minimization ofF to the space where ry,(z), 6v(z) E IP,(T,,) are independent of E for all v. A necessary condition for 1/;h E Wh to minimize F is that the first variation ofF vanishes at 1/Jh for all admissible 'Wh E Wh, that is, (2.7) 15 PAGE 24 For wh E Wh we have Therefore, (2.7) is equivalent to z,, 1 ./ ./ 1-'2 [ + 61 + + 61 <51)] df.'dz + cJ1 + s2 I, + O(s3 ) Zl -1 (2.8) '" 1 = s2 ./ ./ f.'2qb; + aqrJo df.'dz + O(c3), ZJ -1 where ,, 1 I,../ ./ f-'2 [liJ6ry; + l,ry;) + (%<1z + 6,6,) + (i)iryb +iii 0 and for all wh E Wh, in particular for Wh = ,p,, it follows that Thus, z,, 1 0= ./ Zl -1 _, 7 r10 = -u1. dz. ,, Finally, we prove (iii). Because of (ii), we can restrict minimization ofF to the space Wh := {wh E Wh: = -b,(z)}. The choice wh E Wh in (2.7) will zero out the 0(1) and O(s) term in (2.8). Comparing the O(s2 ) term on the left-hand and right-hand sides gives ./,, 2 [(-' -6 ') (-' 6 7 6 ) (7 6' l] 2'' 6' 2 ,_ d 3 + + ry, 2 + u2 2 + CY "1 rJo + TJo 1 + 5"' 1 + a z ,, (2.9) = l q6i + dz 16 PAGE 25 for all ryv E Wh with v 2: 0 and for all Dv E Wh with v > 1. From the choice 6j c= 0, ry0 c= 0, = and 62 = 82, we conclude that ,,, j (if; + 6, )' dz = 0 ==? if; = -6,. (2.10) Substituting (2.10) into (2.9) results in 2 6' 3q 1 + 2aqr}o dz. Choosing 61 = 0, then integration by parts leads to z,, Zr j + 2fr2iforyo dz = 2a j qryo dz, ,, which with (ii) and after division by 2a becomes Jz, 1 d j'' d 3ry0ry0 + aryor}o z = qryo z. (2.11) Because of the choice WhEW", equation (2.11) holds for all ryo E Uh D One major irnplication of Theorem 2.1 is that, when f}v(z) and 6v(z) are continuous piecewise linear functions, (ii) can only be fulfilled if ifo is a linear function. Otherwise, 81 = -TiS is a step function, which would not be continuous. Taking the boundary conditions into account, it fOllows that Tfo:::::: 0. Therefore, U" = {0}, so that (iii) is a vacant statement and does not contradict the fact that fio = 0 is a solution. Consequently) in the diffusion limit c.-+ 0 the discrete minimizer 'h converges to 'h = 0, independent. of the choice of the right hand side q. This shows that the Least-Squares Finite-Element discretization of (1.6) with linear basis elements in space does not give a correct diffusion limit approximation, except in the case q = 0. For a different way of proving this result, we refer to (Manteuffel and Ressel [38]). On the other hand, for piecewise polynomial basis functions of degree 1 > 1, con dition (ii) does not restrict ifo to a linear function. Therefme, Uh contains also nontrivial functions, so that (iii) implies that ?fo is a Galer kin approximation of the clif[usion equation -&4/' +a= q. Thus, the Least-Squares Finite-Element discretization with piecewise poly nomials of degree 1 > 1 yields a correct discrete difrusion limit solution. However, numerical results for a discretization in space by piecewise quadratic basis functions show that applying a scaling transfOrmation (introduced in the next section) prior to the discretization enhances the accuracy. 2.2 Scaling Transformation In this section, we introduce a scaling transformation that is applied to the transport operator prior to the Least-Squares discretization. This scaling transformation plays a key role in this thesis, since it guarantees the accuracy of the Least-Squares discretization in 17 PAGE 26 diffusive regimes even for simple Finite-Element spaces, such as spaces using continuous piecewise linear elements in space (see Section 2.5). To motivate the scaling transformation we introduce the moment reprac;entation of the flux. Let P,(J1) denote the 1-th Legendre polynomial. The normalized Legendre polynomials Pl(Jl.) := J2Y+IP,(Jl.) form an orthonormal basis of L2([-l, 1]): 1 J Pk(Jl.)Pl(Jl.)dJi. = 6kl: -] (2.12) where 8kl denotes Kronecker delta, i.e, 6k1 :::= 1 for k :::= I and 8kl = 0 otherwise. Assuming that 1/!(z, Jl.) E L2([-1, 1]) for all z E [z,, z,], then 1/! has the following expansion (moment representation) in angle: 1/!(z,Jl.) = L . However, the different terms in the operator l, as defined in (1.11), are unbalanced (there are ), 0(1) and O(c) terms), so that different components of the approximation error are weighted differently in The leading terrn of lis P), which means that the error in the higher moments is weighted in this norm very strongly in diffusive regimes (very small c::), even though this part is not important according to the diffusion expansion (1.13). On the contrary, the error in moment zero, which is the important part in diffusive regimes, is hardly measured in the norm since it is weighted by c:. The basic idea is, therefore, to scale equation (1.11), thus changing the weighting in the norm used in the Least-Squares discretization to determine the best approximation to the exact solution in the discrete space. Define for r E m+ the following scaling transformation and its inverse: S := P + r(IP), 1 s-1 = P +-(IP). r (2 15) After applying the scaling transformationS from the left and dividing by s, equation (1.11) becomes 1 1 o PAGE 27 where q, := 5q and 1 a,P 1 Ehf; ,. a,P -51'-= -PI'-+ -(T-P)l'[)z az c [)z Clearly, choosing r = O(c) will increase the weight for moment zero and reduce the weights for the higher mornents. Equation (2.16) can be balanced further by a scaling transformation from the right. Let the domain of operator in (2.16) be the Hilbert space V. Then we define the space V by V:=5-1V, so that (2.17) v = 5-1 v for all v E V and 5v = v for all v E V. Scaling (2.16) also from the rigbt results in 1 a{; r2 .. .. -17/J = ,P = -5!'8-;:;-+-,(I-P)1p + rxP,P = q, [ uz [ (2.18) where 1 l ,-2 -8118 = -(r-r2)(PI' + 11P) + -11I. E E E For r = O(c) we have;' E 0(1), so that in (2.18) tbe derivative of moment zero and one and the moments themselves are weighted equally. Moreover, we point out that the double-scaled operator ,8 can be bounded independent of s. In the Least-Squares context, the additional scaling from the right can be avoided, smce ( ;[;-q,, ;[;-q,) ,PEV (2.19) = min (/J-q, ,/J-q,), >tEV which will simplify the boundary conditions and so also the computations. Further, for slab geometry, because of transformation (2.1) we may assume witlwut loss of generality that CTt and parameters are constant in space. However, fOr higher di1uensional problems, this cannot beAestablished, so that s :::: s(r). For inhomogeneous rnaterial, s(r) is in general discontinuous, so that the scaling parameter r, which was chosen to be O(s), would be discontinuous. To perform the scaling would then require to prescribe jump conditions in the scaled solution V across material interfaces. Therefore, we use the additional scaling from the right only as motivation for the choice of the scaling transformation and as a tool in the theory in Section 2.4, where we exploit the nice form of the double scaled operator (2.18). For another way of motivating the scaling by way of the moment equations, we refer the reader to Manteufl'el and Ressel [38]. As outlined in Section 1.4, a necessary condition for '1/J E V to be a minirnizer of the Least-Squares functional (2.19) is that the first variation vanishes at 1j;, which results in the problem: find ,P E V such tbat a( ,P, v) := (,P,v) = (q, ,[.v) \1 v E V. (2.20) For a discretization of problem (2.20), the bilinear form a(-,) is restricted to a finite dimensional subspace Vh C V. In the remaining of this chapter, we analyze the error of this discretization for various subspaces yh. 19 PAGE 28 2.3 Error Bounds for Nondiffusive Regimes In this section we establish bounds in an unsealed norm for the discretization error of the Least-Squares discretization. However, in this norm it is not possible to prove V ellipticity and continuity of the bilinear form (2.20) with constants independent of parameters and a. In diffusive regimes, where is very small, these bounds blow up and are therefore useless. Nevertheless, the bounds for diffusive regimes that are derived in Section 2.5 are only valid for < 1/v'3, so that the bounds in this section can be used to cover the range [ 2: 1/v'3. As outlined in Section 1.4, the first step on the way to bounding the error is to prove V-ellipticity and continuity of the bilinear form aC, ) in some norm. From the view of standard elliptic boundary value problems, the choice V = H1([z,, z,]) x L2([-1, 1]) (Adams [1]) with the norm 2 ;'';1 (8v)2 2 llvlh_o := [}z + v dpdz Z! -1 seems natural. However) it is easy to see that the bilinear form a(,) cannot be bounded from below in this norm. Let Vk := V2 sin(brz) B(p) with for I' E [-8, 0] for p E [0, 8] otherwise Then, for all k E IN, we have Vk E H1([z,,z,]) X L2([-1, 1]) and llvklh.o = (k7r)2 + l. Some simple calculations show that a(v, v) <; Choosing 8 = -k1 then the bilinear form a(-,) is bounded for all k while lim llvklko = oo. 7r k-HXJ Thus, there is no lower bound for a(-,) in the norm 111 0 The next obvious choice is II [} 112 2 v 2 lllvlll := I' [}z + llvll (2.21) Closure here is with respect to the norm 111, so that Vis a Hilbert space. In the following, we bound the Least-Squares discretization error in norm (2.21) for various Finite-Element spaces. 2.3.1 Continuity and V-ellipticity Before we establish continuity and V-ellipticity of the bilinear form a(,), we summarize some simple properties of our operators. 20 PAGE 29 Lemma 2.2 (Properties of P, Sand pffz) For all 'UJ v E V 1 we have: (i) (Pu, v) = (u, Pv); and ((IP)u, v) = (u, (IP)v); P2 = P; and (IP)2 = (IP). Thus P and (I P) are orthogonal projections; (ii) (Pu, v) = (Pu, Pv); and ((IP)u, v) = ((IP)u, (IP)1;); (iii) II vii:<: II for scaling parameter r = 6 and f :<: 1. (iv) IIPvll' ::0: ( IIPvii-IIP(I-P)vll) 2 ; (v) v) ::0: 0; v z Proof. (i): '' 1 Zr 1 (Pu, v) = J J Pu v dpdz J Pu I v dpdz Zi -1 Z! -1 z ,. z ,. 1 I 2Pu Pv dz = I Pv I u dpdz Zl Z/ -1 '' 1 = I I u Pv dpdz = (u, Pv), Zt -1 and the second identity follows directly from the first. From the definition of P, it is obvious that P2 = P and, therefore, (IPj2 = (IP). (ii): follows immediately from (i). (iii): llvll2 = IIPvW + II(I-P)vll' :<: ;\11Pvll2 +II (IP)vll' = II;Svll', since 6 :<: 1. (iv): ,, l llpvll' =.I .I p2 [Pv +(IP)v]2 dp dz Z/ -1 1 2 j'' /1 2 2 = 3IIPvll + 2 p Pv (IP)v dp dz +lip( IP)vll Zj -] 21 PAGE 30 The mixed term can be bounded by the Holder inequality as follows: ,_ 1 ,_ 1 j j 112Pv (IP)vdpdz :S j IPvl j f1 [fli(I-P)v] d11 dz Zl -1 ZJ -1 :S JI U 1Pvl2 dz) 112 U j lf1(I-P)l/'12 dp dz) 112 1 :S y3 IIPvll llfl(I-P)vll Therefore 2 1 2 2 . 2 llflvll ?: jj11Pvll y3 IIPvllllfl(l-P)vll + llfl(I-P)vll = IIPvll-llfl(I-P)vll) 2 (v): Applying integration by parts with respect to z, we get z,.l 1 Zrl j j f1 v dpdz = j f1 [v2(ze: f!)-v2(z,, 11)] dp/ j f! v dpdz. Zj -1 -1 Zl -1 Taking into account the boundary conditions for v E V 1 it follows that 1 (fl = j f1 [v2(z, fl)-v2(z,, p)) dp -1 0 It now easily follows from the Cauchy-Schwarz inequality that the bilinear form aC, )is continuous in the norm 111, since for any u, u E V ia(u, v)l = II(.Cu,[v)l :S II.Cuiiii.Cvll Here C, := [ ( 2 + ( )'], and we used the discrete Holder inequality in the last step. 22 PAGE 31 We prove now V-ellipticity of the bilinear form a(-,) wben a =F 0. Lemma 2.3 (V-ellipticity for "'=F 0) Suppose o =F 0 and let T = .,ffr. Tben there exists C, > 0 such that, for all v E V, a(v, v) 2: C, lllvlll2 whereCe:=min{f;,a, a2}, Proof. We have a(v,v) (J:.v, J:.v) 1 II iJv II' II iJv II' c' Pp.iJz +a (I-P)p.az + "' II (IP)vll2 + "'2IIPvll' (2.22) +P11-, Pv +-(IP)p.-, (IP)v 2o \ iJv ) 2"' \ iJv ) [ 8z s az The second mixed term can be written using (ii.) of Lernrna 2.2 as According to (v) of Lemma 2.2, the first term here is always positive and the second tern1 cancels with the first mixed term in (2.22), so that a(v, v) = :,IIPp. r +ct II (Ir + "' II(!-P)vll' + a'IIPvll' > C, lllvlll', with Ce := min o:2}, which proves the lemma. 0 For the more difficult task of establishing the V -ellipticity of the bilinear form a(, ) when a= 0, we need the following PoincarC-Friedrichs inequality. Lemma 2.4 (Poincare-EHedrichs Inequality) For any v E V, we have Proof. We have iJv(s, p.) f1 iJs 23 (2.23) for f1 2: 0 = for f1 :S 0 PAGE 32 Jz I" &v(&ss, I') I ds r for I' 2: 0 ,, [!'v( z, p.) I < I ds for I':; 0 z Taking into account that assumption (zr-zi) = 1 implies we obtain the lemma. D We are now in a position to establish Lemma 2.5 (V-ellipticity for a= D) Suppose that a-0 and 0 :S s :S 1, and let Then there exists Ce > 0 such that, for any v E V, a(v, v) 2: C, [llv[[[2 where Ce = Proof. Recall that 1 &v r &v r Lv = E Pf' &z + E(I-P)p &z + (IP)v + rxPv. Because of (i) in Lemma 2.2, we have a(v,v) = (Lv,Lv) r2 . 2r2 I &v ) + s4 ((IP)v, (1 -P)v) + $$IP)p az, (IP)v Analyzing the mixed term and using (i) of Lernma 2.2, we see that I av ) 1 av ) 1 av ) 1 av ) \(IP)p0z, (IP)v = \(IP)t' &z, v = \'' EJz, v -\ Pf' EJz, Pv 24 PAGE 33 The first term is always positive according to (v) of Lemma 2.2. Consider the following arithmetic-geometric inequality: for any r; Em+ and for any a, bE ill, 2ab < 7Ja2 +!C. We can thus bound the second term according to Therefore, the bilinear form a(-, ) can be bounded from below by Defming T2 T2 +-.11(1-P)vll2 -siiPvll' E E 7] 0 -IIPvll2 .llvll' IIPI'8 II' ,._ az II' so that so that ( 1 _b)= II (IP)vll2 II vii' (1 ) = liU--r the above bound thus simplifies to a(v,v) + C, llvll', with cl = 2. (-r-72 rn+r2(1--r)), ,;2 c c, = 2. (r' (16)-72 o) c:2 s2 cr; (2.24) By proper choice of q, we now need only establish that C1 and C2 are positive. Unfortunately, for large enough 6, C2 will be negative, so we will need in this case to readjust the terms in (2.24), which we do by way of the PoincareFriedrichs inequality. Case 1: 6 > Jj: From the PoincareFriedrichs inequality (2.23) and (iii) of Lemma 2.2, we conclude that II' 2: lll'vll2 2: IIPvii-IW-P)vll) 2 Since I' E [-1, 1], tben clearly Ill'( IP)vll :'::II (IP)vll Therefore, 1 1 y'SIIPvll-Ill'( IP)rll 2: y'SIIPvll-II( IP)vll > 0, where the last inequality follows from the assumption 0 ?:: g > i since 1 2 .. 2 1 3 3IIPvll 2: 11(1 P)vll <= 3b 2 (16) <= 6 2 4 25 (2.25) PAGE 34 From (2.25), we get ( 1 )' ( 1 )2 0)11Pvii-IIJ.l(l-P)vll :?: 0)11Pvll-II( I-P)vll It then follows that II' :?: ( IIPviHI(IP)vll)' = v'0=TJ)' llvll' 1 12 1 2 ( 1!;)2 :?: 0) /fl13 II vii Thus, II 8vll' 1 2 J.l (jz :?: 13 llvll We now use (2.26) to rewrite (2.24) as a(v, v) :?: ( C,-;;, ) II' + II' + C, llvll2 :?: ( C,+ ;;, ) II' + + C,) llvll2 Choosing 17 ::::::: 5 } and using the fact that 6 ::=; 1 results in 1 ( r2 2 ( 1 1 ) ) r2 = c' (1 -li) + 7 26 52 :?: 52r2 > O, and c,-;;, = 1 (1 (1 _7 (1+ !;)) + r;) > r;, since r = 1/}2+ ;;, so r2 (1+ 5 2 ) < 1. Case 2: 6 < H= Choosing TJ = 24s, for cl and c, in (2.24) we obtain that and cl = 12 (1(1-24r2 ) +r2(1-1)):?: 2 > 0, E since 24r2 ::::::: < 1 for s < fi. v 22 26 (2.26) PAGE 35 Thus, altogether we have a(v, v) ::0: C, lllvlll2 with =-mm --1-->-mln --C r2 { 1 1 1 } 1 { 1 1 } e 521 21 1 262 54 52 1 26 2 1 which completes the proof. D From continuity and V -ellipticity of the bilinear form a(-, ) it follows directly from the Lax-Milgram Lemma (see Section 1.4) that problem (2.20) and all of its discretizations are well posed. The next step is to obtain discretization error bounds for a variety of discrete subspaces vh' which is done in the next subsection. 2.3.2 Error Bounds As outlined in the introduction, continuity and V-ellipticity of the bilinear form a(-,) lead directly to Cea's Lemma (1.24). Therefore, bounding the discretization error 1111/> -1/>hlll is reduced to the problem of bounding min 1111/>-vhlll, which is a problem of approximation theory and depends on the choice of the finite-dimensional space Vh. Here we consider two main classes of discrete spaces vh. The first consists of spaces with functions that can be expanded into the first normalized Legendre polynomials with respect to the direction angle f-1 and are piecewise polynomials of degree r in zona partition Th of the slab [z1, Zr]. This choice of the finite dimensional space V h corresponds to discretization by a spectral method in angle jJ and a Finite-Element discretization in space. In transport theory the spectral discretization in angle with the first N Legendre Polynomials as basis functions is also called a PN-1 discretization in angle. For any f(z) E Hm([z1, zr]), with 1 <; m <; r+ 1, let IT,f(z) denote the interpolant of f(z) by piecewise polynomials of degree r 2 1 on a partition of [z,, z,]. It then can be shown (Jobnson [25, p. 91]) that llf(z) -II,f(z)llv([' PAGE 36 Then the error of the truncated expansion can be bounded as follows Lemma 2.6 (Truncated expansion into Legendre polynomials) For r?: 0, let g(z, p) E H"([zr, z,]) x H2([-1, 1]) and let llN be defined as in (2.28). Then have: (i) IIIINgll :::; llgll; (ii) llim)(z)ll :'0 r(l!l) \lm:'O rand \lz E [z,,z,]; (iii) For any m r, we have c II amg II < N .Cs azm (2.30) with C independent of g and N. Proof. (i): liN is orthogonal projection with respect to the inner product of 2([-1, 1]), so IIIINgll'([-1,1]) :'0 llgl!p([-1,1]) and hence lll1Ngll :'0 llgll (ii): By definition (2.29) and integration by parts, we have 1 j1 am9 1 II Ergll = 21(1 + 1) .Cs azm PI(!') dp :'0 V'il(l + 1) .Cs azm L'([-1 1]). -] (2.31) Therefore, < 1 ll.c amg II "'1 I(/+ 1) 8 azm (iii): Since the Legendre Polynomials are an orthogonal basis, from (2.31) we obtain ___-II _g = 2 I: 1(m)(zW < .C ___ .Z::: . llam am II' oo II am II' oo 1 i)zm N azm L'([-1,1]) I=N I -S iJzm L'([-1,1]) l=N [/(/ + 1)]2 For l 2: 1 we have f so that the sum can be bounded by 00 1 I: [/(/ + 1 )]' I=N 00 00 1 00 1 ;1 4 4 :'0 4 I: (I+ 1)4 = 4 I: /4 :'0 4 /4dl = 3Ns :'0 3N2' I=N I=N+l N Therefore with C = .ji, which proves the lernrna. 0 28 PAGE 37 Theorem 2.7 (Finite-Element in space, spectral discretization in angle) Suppose that Th is a partition of the slab [zr, Zr] with maximum mesh size h. _Let V be given as defined in (2.21). Define N-1 Vh = L E JP,(T,,) for I= 0, ... ,N -I where 1P r (Th) denotes the space of piecewise polynomials of degree :::; r on the partition Th. Suppose 1 :S m :S r + 1 and let 1/; E V n (W"([z,, z,]) x H2([ -1, 1])) be the solution of (2.20) and 1/Jh E Vh be the solution of (2.20) restricted to Vh Then Proof. From Cea's Lemma (1.24), we have Now note that lllvlll :S llvll1 0 Therefore, by (i) ofLemma2.6, (2.27) and (2.30), we conclude < cl 1' 'II + c hm-l N 1 soc 1,0 2 which proves the theorem. 0 The second n1ain class of finite-dimensional spaces considered here are formed by functions that are piecewise polynomials in space z as well as in angle J-.L This choice corresponds to a FiniteElement discretization in both space and angle with rectangular elernents. Suppose that Th is a partition of the computational domain D = [z,, z,] x [ -1, 1] into rectangles T = [z;, Z;+J] X [l'v, l'v+!] of maximum diameter h. To be able to handle the boundary conditions properly, we assume in addition that (.z:1, 0) and ( Zr, 0) are nodes of the triangulation 'TJ,. By Th we deJine the discrete space: vhiT= L 'iTETh o:;f3,r'!:.r 29 (2.32) PAGE 38 For all v E V, let IIJ,v E V" denote an interpolant1 of v with respect to the partition T,. It can be proved (Ciarlet [16, Theorem 16.2]) that, for v E V n H"+1(D) the following bound for the interpolation error holds: where 0 :'0 rn :'0 k + 1 and llw+'(D) is the semi-norm of Hk+1(D) (Adams [1]). Combining Cea's Lemma and (2.33), we get Theorem 2.8 (Finite-Elements in space and angle) Let V", T,, h be given as defined above. Suppose 1/J E V n H"+1(D) is the solution of (2.20) and let 1j;, E 11" be the solution of (2.20) restricted to V" defmed in (2.32). Then we have: Proof. By Cea's Lernrna, we need only to bound llhD-lh1f;lllNote that, for all v E v n w+1(D), lllvlll :'0 :'0 llviiH'(D)' Thus, using (2.33) with m = 1, it follows that 1111/J-II,1/JIII :'0 111/J-II,1/JIIH'(DJ :'0 C h" 11/Jiw+>(Dl which proves the theorem. D We point out that the error bounds in Theorem 2.7 and Theorem 2.8 depend on the ratio In the V -ellipticity bounds in Lemma 2.3 and Lemma 2.5, the scaling parameter Twas chosen to be O(s). Therefore, when is small, C, is O(a) when a f 0, while C, is 0(1) when o = 0. In addition, for T = 0(), the continuity constant C, is 0(-;), so we which blows up for diffusive regimes, where is very srnall. However, numerical results show that the Least-Squares discretization of the scaled transport equation stays accurate in diffusive regimes. Thus, we conclude that the bounds, derived in this section, are not sharp enough to reflect the aceuracy of the Least-Squares discretization in diffusive regimes. In order to obtain error bounds that do not blow up in diffusive regirnes, it is essential to prove continuity and V -ellipticity of the bilinear form aC )with constants independent of parameters and a. This is done in the next section with respect to a scaled norm. 2.4 Continuity and V-ellipticity with respect to a scaled norm In this section, which is the central part of this thesis, we prove continuity and 11-ellipticity of the form a(-,) in (2.20) with constants independent of parameters f and a. This is the foundation for the bounds in Section 2.5 of the Least-Squares discretization error that do not blow up for diffusive regimes. Throughout this section, we assume that r = f and a:::; 1. In order to obtain continuity and V -ellipticity with constants independent of c and a, we use a scaled norm. To motivate its choice, we look at the double-scaled (from left and right) r 2: 2, there are many different interpolants, depending on the choice of Lhe support absci!lsas and support ordinates on the rectangle, which are not specified here. For an overview of counnonly used lnterpolants for rectangles, we refer the reader to Ciar1eL [16, p. 129]. 30 PAGE 39 transport operator (2.18). Let V denote the domain of the single-scaled (only from the left) transport operator (2.16) and v = s-'v the domain of the double-scaled transport operator (2.18). Defining we see that the norrn 1 Q := -SpS = (1c) (Pp + pP) + Epl, [ 2 v 2 II [) llvllv := Q [)z + llvll for V E V would be a natural choice for bounding the double scaled bilinear form a(u, v) := (Su, SV). (2.34) (2.35) (2.36) However, because of the reasons mentioned in Section 2.2, it is desired to use the single scaled transport operator for the computations. Therefore, u."ling the relation V = s-1 V, we derive from (2.35) the following norm for v E V: llvllt = + 11"112 = \\ 1\' + IIS-1vll2 (2.37) = + IIPv+ P)vll' = II r + II P)vll 2 + IIPvll' = llvllv. We define v-v(z,,p)=Oforp>O; v(z,,p)=Ofor!i. PAGE 40 Employing the discrete HOlder inequality and using the assumption o :::; 1, we obtain !!Cull II + II P)ull +!!Pull V3 + PJul!' + IIPull'r' =V311ullv Thus, for all u, v E V, la(u, v)l C, llullv llvllv with C, = 3. To prove V-ellipticity of the bilinear from a(.,), we exploit the convenient form of the double-scaled transport operator and prove first that the double-scaled bilinear from a(-,) in (2.36) is If-elliptic. The V-ellipticity of the bilinear from a(,) then follows easily as in Corollary 2.12. In order to prove V-ellipticity of the bilinear from a(-,), we need the following lemmas. Lemma 2.9 For all u)v E V, V E V and c: :S 1, we have (i) rll(l-P)i!ll llflvll; (ii) ( ?_ 0; (iii) (Poineare-Friedriehs inequality) Proof. (i): Since Ill"( IP)vll II(I-P)vll and then ,, 1 Zr 1 Zr j j p2 (Pv)2 dpdz = j (Pv)2 / p2 dpdz = j (Pv)2 dz ZJ -1 ZJ -1 Zl ,, 1 = J // (Pi!)2 dpdz = jiiPi!ll', Zj -1 ll11v11 = ll!t[P + r(I-P)Jvll ?_ lli"Pvl!ciii"(I-P)vll ?_ }ei!Pi!llcii(I-P)i!ll (2.39) (ii): From (i) in Lemma 2.2, it follows that (Su, v) = (u, Sv) Vu, v E V. Therefore, using (2.17) leads to IQ!Jv l1s si!v 1 I sav 1 I i!v ) o \ 8z'v = \;; f-1 8z'v = ;;\J-L 8z'Sv = ;;\f-la;,v where the last inequality follows from (v) of Lemma 2.2. 32 PAGE 41 (iii): From the PoincareFriedrichs inequality (2.23) proved in Lemma 2.4, we have llpvll :S II Using (iii) of Lemma 2.2 and the relation (2.17) results in 0 The following technical lemma is tedious but simplifies the proof of the major result, Theorem 2.11. Lemma 2.10 Suppose 0 :S E :S Js Then, for any bE [0, 1], there exists b 2 0 such that H(b,o) = { J(o(1 + jJ&)' + 41i(1-li) + (1-i) &H} < 0988, where s := [o1i2 -cv'3(1-li)1i2 ] 2 In particular, for li < 0.875, we can choose b = 0. Proof. For /j < 0.875, we choose b = 0 and get H(O, li) = { v/4.5-3.52 + o} :S H(O, 0.875) < 0.986, since H(O, 6) is monotone increasing forb E [0, 1]. For 6 > 0.875) using the assumption c:v'3 < 1, we have Suppose we restrict the choice of b to b :S It then follows that since s :S 1. From (2.40), we conclude that Therefore, Simple calculus shows that 3.5 b* := (3 + !3) (3-fJ) < (3 + (i) v --p 4 33 (2.40) PAGE 42 rr1inirnizes ii and that b* > 0 if 8 > 0.875. After tedious but straightforward rnanipulation1 we have that Ti(b*, 6) attains its maximum at 6*"' 0.893 and that H(b*, 6*) :S 0.988. D We are now ready to prove the central result of this section. Theorem 2.11 (V-ellipticity of a(-,)) Let a(-,) and II llv be given as in (2.36) and (2.35). Suppose that 0 :Sa :S 1, 0 :Sf< :To Then there exists a constant Ce > 0 such that, for all V E V, a(ii, ii) = + aPii +(I-P)iill' ?: c, + 11"11') = c, 11"11&. where C, = 0.012, which is independent of s and a. Proof. We have a(ii, ii) + aPv +(I-P)iill' + a'IIPvll' + IIU-P)iill' +2a \ + 2\ (I-P)v) = + a2IIPvll2 + II(I-P)vll2 +2a: \ + 2(1-a)\ (I-P)ii). (2.41) (2.42) For the last term, we may write for any d E [0, 1], by using (ii) of Lemma 2.2 and (ii) of Lemma 2.9: > -d \ PQ Pv) + (1-d)\ (I-P)Q P)v). Substituting this into (2.42) and bounding the fourth term in (2.42) by (ii) of Lemma 2.9, 34 PAGE 43 we get a(v, v) > IIQ 112 + a'IIPilll' +II (I-P)vll2 -2(1 -")d 1 (PQ Pv) 1 2(1 -")(I d) 1 ((I -P)Q g:, (I-P)v) 1 which can be reduced by setting a = 0 to Defming a(v, v) ::: IIQ g: I!'+ 11(1-P)vll' -2d )(PQg:, Pv))-2(1d) )((I-(1-P)v)). 0 -IIPilll2 llilll' IIPQ*II' '!-)IQB'jl2 1 Dz I so that so that ( 1 o) = II (I-P)illl' llvll' 1--II (I-P)Q*II' ( r)-))Qg:ll' and using Cauchy-Schwarz inequality, we conclude from (2.43) that a(v, v) ::: IIQ (1-6)11"11'-2dVt.,f'i IIQ llllilll -2(1-r) (2.43) (2.44) To maximize the lower bound in (2.44), we divide the region (6, r) E [0, 1] x [0, 1] into two triangles and choose d as follows: d { 1 for o + /' :S 1 0 for o + /' ?: 1 Next) we consider these two cases separately. Case 1: o + /' :S 1, d = 1: For any rJ > 0 and any u, v and any norm II II, the arithmetic-geometric mean inequality holds. We thus have 1 2llullllvll < ryllull' + -llvll2 ry II Er II' ( o) a(v, v)::: (1-rry) IQ + 1-o-I Iilii' (2.45) It remains to choose 1) such that the terms (1-rrJ) and ( 1-oin (2.45) can be bounded by a positive constant from below for all possible/', o with/'+ o :S 1. Foro < 0.5, we choose 1) so that (1-r1J)= 35 PAGE 44 which yields b + vo2 +40, ,, = 2-y Applying Lemma 2.10, since -y + o :S 1 and therefore, -y :S 1o, then we have Thus, from (2.45), the V-ellipticity of a(-,) directly follows with c, 2: 0.012. On the other hand, if 8 2: 0.5, the second term in (2.45) can become negative. To keep it positive, we then rewrite (2.45) for any bE [0, 1] as follows Since o 2:0.5 and E < 1/\1'3 we have 0 :S (j,v'b-cv'l=6). We can now use the Poineare Friedrichs inequality (2.39) of Lemma 2.9 and inequality (i) of Lemma 2.9 to bound the second term by which results in (2.46) where s = (Vi-c!3J0=bl) 2 (2.47) Again, we choose rJ so that which yields -(b + s 8) + J (b + s 6) 2 + 40, 2-y Next, to attain a positive constant in the lower bound in .(2.46), we need to show that for all possible 6, 'I with 8 + -y :S 1, b 2: 0.5, a positive b can be selected so that Since G1(b,l5,-y) :S G1(b,6,1-15) for 8+-y :S 1, it then is sufficient to prove V6E[0.5,1]3b>0: G1(b,6,1-li) PAGE 45 Case 2: 6 + 7 2: 1: Setting d = 0 in (2.44) and preceding as in case 1 results in a(11,11)2:(1-(1-7)'7) +(1-6-1 ;6 ) 111111 (2.48) For D :::; 0.5, we choose 6 + J6' + 4(16)(1-7 ) rJ= 2(1-7) so that (1-(17)'7) = ( 1b-1 ; 8). Using Lemma 2.10, since 6 + 1 '2: 1, and therefore, 6 '2: 1-/, we then have (17)'7 = { J62 + 4(16)(1-7 ) + 6} s { J62 + 4(16)6 + 15} = H(o, 15) s o 988. The V-ellipticity of a(-,) then follows with C, = 1-0.988 = 0.012. On the other hand, for 8 2: 0.5, we introduce, as in case 1, a parameter b E [0, 1] and use the PoincareFriedrichs inequality (2.39) to conclude from (2.48) that a(11,11) 2: (1-b-(1-7)'7) + (1-8+js-1 ;8 ) 111111, (2.49) with s as defined in (2.47). Again, we fi.rst choose ry, so that (1-b-(17)'7) = (18 + 1 b$$ 3 1) which yields (b+ .5) + J (b+ &s-8) 2 + 4(16)(1-7) 1)= 2(1-7) In order to attain a positive constant in the lower bound (2.49), we need to show that, for all possible 6, 7 with 8 + 7 2: 1, 6 2: 0.5, a positive b can be selected so that, G,(b, o, 7) := b + (17)'7 = q J (b+ js-0 r + 4(1.5)(1-1) + (b +6-H} < C* <1 Since G,(b, .5, 'I) S G,(b, .5, 1-b) = H(b, 8) for 8+7 2: 1, this follows directly from Lemma 2.10 with C* = 0.988. Finally, from (2.49) with c, := 1C* = 0.012 the V-ellipticity of a(,) follows, which proves the theorem. 0 From the V-ellipticity of the bilinear form a(-,) the V-ellipticity of the bilinear form a(-,) can be proved as follows. Corollary 2.12 ( V -ellipticity of a(-,) ) Let a(,) and llllv be given as in (2.20) and (2.37). Assume that 0 Sa S 1, 0 Sf< Js Then there exists a constant Ce > 0 such that, for all v E V, a(v, v) 2: C, (2.50) 37 PAGE 46 where Ce = 0.012, which is independent of a and c. Proof. By the definition ofthe norm llllv in (2.37) and the relation (2.17) we have llvllv = llvllvTherefore, using (2.41) in Theorem 2.11 we obtain for any v E V Zr 1 Zr 1 a(v,v) j j .Cv .Cv dfldz = j j .CSv .CSv dfldz ZJ -1 Zj -1 which proves the corollary D 2.5 Error Bounds for Diffusive Regimes Using continuity and V-ellipticity of the bilinear from a(,) in the norm llllv with constants independent of ex and c) in the following section we establish discretization error bounds that do not blow up in the limit c _,. 0. we use the same discrete spaces introduced in Section 2.3.2. We first consider discrete spaces with functions that can be expanded into the first N normalized Legendre polynomials with respect to the direction angle fl (PN-1 discretiza tion in angle) and are piecewise polynomials of degree ::; r in zona partition Th of the slab. To combine Cea's Lernrna (1.24) with the interpolation error bounds in (2.27) and (2.30) to obtain an error bound for this class of discrete spaces, we need the following le1nma. Lemma 2.13 (Bound for commutator [IIN.C-.CIIN]) Let .C be the transport operator as defined in (2.16), .Cs the Sturm-Liouville operator as defined in (2.29) and liN the projection operator as defined in (2.28). Suppose N 2: 2, m 2: 0, and v E V n Hm+3(D). Then, there exists a constant C > 0 independent of c and a such that Proof. Recall that 1 a a 1 .C =-PI"-+ (IP)l"+-(IP) +a P, c Dz Dz E and that has the moment expansion (see (2.13) in Section 2.2) Note that and D"'v ) ( ) azm -L 'f'l .z Pr J.L 1:::::0 with I ) }' amv(z,fl) ( ). d 'PI z -2 iJzm PI I" I" -I 38 (2.51) PAGE 47 Therefore) i) i) [liN [-[ IIN] =liN I" {)z-I" IIN (jz" Using the relation (see Chapter 4) with we see that bl := l + 1 d ( ) 0 an P-1 fJ = y'4(1 + 1)'1 a amv IIN __ r &z &zm N "' _,c(m+I) b = L '+'/ l-1PI-1 1::::::0 N-2 + I: fm+1 ) b1PI+1 /::::::0 On the other hand, N-1 =I" I: )m+1 ) PI 1::::::0 N-1 N-1 "' = L., '+'I b1-1PI-1 + I: fm+l) b, PI+! /::::::0 1::::::0 Thus, in combination with (2.52) we have b (AC(m+l) _,c(m+1) ) = N-1 '+'N PN-1 -'+'N-1 PN Now we notice that) for any integers k) l ?: 0) z,. l Z.r llfm+1 lPkll' = / (fm+Jl(z)r / 2/ (fm+ll(z))" dz ;q -l Zt = llr/>fm+l)ll' :'::: 1 [1(1 + l)]' II 0m+lv II' [s iJzm+1 (2.52) (2.53) where the last inequality follows from (ii) of Lemma 2.6. Therefore, (2.53) can be bounded 39 PAGE 48 as follows: smce < 8 1 3 N2 {::} 1 1 8 1 <--,13 3 N2 II [jffi+l 0 vh(z,.,p) = 0 for I'< 0}, where lPr(Th) denotes the space of piecewise polynomials of degree::; ron the partition Th. Suppose 0 :S E < Ja. 0 :Sa :S 1 and 1 :S rn :S r+ I. Let 1j; E VnHm+3(D) be the solution of (2.20) with right-hand side q, E Hm+2(D). Let 1/Jh E V" be the solution of (2.20) restricted to Vh Then < (II 8(1/; 1/Jh) 112 +II P)( 1j; -1/Jhf + IIP(' 8(1/; 1/Jh)) II :S seh, II (IP) (fl8(1j; 1/Jh)) II :S eh, (2.54) Proof. Using Cea's Lemma (1 .24) and the V-ellipticity of the bilinear form a(-,), from 40 PAGE 49 Corollary 2.12 we conclude that since fi,IIN'JI E Vh In order to bound the first term in (2.55), we use (2.51) of Lemma 2.13 and (2.30) of Lemma 2.6 to get IIL PAGE 50 Finally, substituting (2.56) and (2.57) into (2.55) results in (2.54). D Remark 2.15 (Interpretation of error bound) In the following, we interpret the error bound (2.54) more closely. For diffusive regirnes (c < 1), the exact solution of the continuous problem has the diffusion expansion2 (see (1.13) in Section 1.2.2) 1/J(z, f.!)= ;(z) + E;R(z, f.!), while the the right-hand side in this case is assumed to have the form q = qo(z) + EMI(z) + O(s2). Therefore, it follows that Csl/J = O(s) and Csq, = O(t:2), since q_, = 8q = Pq + r(I-P)q. Taking this into account, for the error eh in (2.54) we get 1 (C1 2 C2 ) 1 ( '"llfJ'"q,ll hm l) e" = VG; NO(s ) + N' O(s) + C, C3h f!zm + C4 N' O(s Thus, the error in the zeroth rnoment [[P( 1j; -1/!h)[[ is bounded by 0( h'" )+O(s) and the error in the higher moments [[(IP)( -1/Jh)[[ is bounded by O(sh'") + O(c2). In partieular, for diffusive regimes, where c is very small, convergence of the discrete solution is also assured by the above bound for small N, which is a reasonable choice in this case, since the exact solution is nearly independent of ft. Moreover, the bound in 'rheorern 2.14 directly gives the optimal order of conver gence lor the spatial discretization without the use of Nitsche's trick (Johnson [25, p. 97]). For example, for piecewise linear elements, which means r = 1, we can choose m = 2 and get an O(h2 ) error bound in the L2-norrn under the regularity assumptions on 1./; required in the theorem. On the other hand, if c: is close to )3, so that E > hm and E > 1/ N, then an error bound can be obtained more easily, since Therefore, for any v E V n (Hm+1([zr, Zr] x H2([ -1, 1]), Cea's Lemma (1.24) and the bounds in (2.27) and (2.30) for the interpolation error lead directly to rc: ( 1 \ 1/2 ( c II 8'"1/J II ) :S V Cc l + c2) )r [[Cs PAGE 51 l''or the case c > Js, which is not covered by Theorem 2 .14, the error bounds in Section 2.3.2 can be used instead. D The second main class of finite-dimensional spaces considered here are formed by functions that are piecewise polynomials in space z as well as in angle fl. This choice corresponds to a Finite-Element discretization in both space and angle. Because Section 2.3.2 contains error bounds for this class of discrete spaces that are valid in non-diffusive regimes, we concentrate in the following only on error bounds for diffusive regimes. TherefOre, we assume in the following theorern, which combines Cea's Lemma and (2.33), that the exact solution ha..-; a diffusion expansion, which sim.plifies the proof. Theorem 2.16 (Finite-Elements in space and angle) Suppose that 0 :S a :S 1 and 0 :S c :S }3. Let Th be a partition of the computational domain D = [z,,z,] x [-1,1] into rectangles T= [z;,z;+J] X [!'v,!'v+t] of maximum diameter h. To be able to handle the boundary conditions properly, we assume in addition that ( z1, 0) and (zc, 0) are nodes of the triangulation. Let V be given as in (2.38) and define := { v, E C0(D); vhiT= :z.::: 'ITETh o:::;fJ,r:::;1 vh(z,,rt)=O for rt>O, vh(z,,rt)=O for rtR(z,rt) is valid. Then we have 111/J-q,,liv :S [ C h' (11/Jiw+(D) + 1Riw+(D)), with C independent of c, a, and h. Proof. From Cea's Lemma (1.24), we have 111/J -1/Jhllv :S [ 111/J -llh'-1lhli>)11). 43 (2 59) (2.60) PAGE 52 By (2.33) we now bound any of the above four terms separately and use the fact that IIh is as an interpolation operator linear, so that llhV' = IIh + sfihR Since (z) in the diffusion expansion of 1/J is independent of angle fl, we conclude that IIh(z) is also independent of I' Therefore, Pp'Jt; = 0 = PI'%Jlh, so that (2.61) where the last inequality follows from (2.3:l). Using (2.33) for the sec.ond term in (2.60) results in (2.62) Because and IIh are independent of angle, we have (IP) = 0 = (IP)Uh Therefore, the third term in (2.60) can be bounded by (2.63) since h < (z,-z,) = 1. Similarly, a bound for the last term in (2.60) is given by (2.64) Inserting (2.61), (2.62), (2.63), and (2.64) into (2.60) results in (2.59), whicb proves tbe theorem. 0 Remark 2.17 (V-ellipticity constant C,) Both error bounds (2.54) and (2.59) depend on the reciprocal of the V-ellipticity constant C,. According to Theorem 2.11 and Corollary 2.12, C, = 0.012, so 1/C, = 83.3, which is fairly large. However, we would like to point out that we simplifted the proof of Theorem 2.11 by considering only the worst case a= 0. Without setting a= 0, (2.46) would change to a(v, v) ;;: \\Q (1o) 11"112d(1a)Jbvr \\Q -2(1-d)(l(2.6.5) which clearly shows that the V -ellipticity constant Ce. increases with a. 'To judge the quantitative behavior, we computed Ce for certain values of a using (2.65). The results are plotted in Figure 2.1. Already for a= 0.3, 1/C, drops down to 7.04. D 44 PAGE 53 Ce vs. alpha i 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 /Ce vs. alpha 100 8 60 40 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 2.1: V-ellipticity constant C'e and its reciprocal as a function of the absorption parameter a. 45 PAGE 54 CHAPTER 3 X-Y-Z GEOMETRY In this chapter, we generalize the scaling transfOrmation and the error bounds for the Least-Squares Finite-'Elernent discretization, from one-dimensional slab geometry to three dimensions. Since the main focus of this chapter is on diffusive transport problems, in the following we use the parameterized form (1.14) of the transport operator l. In addition, we assume that the total cross section O't is constant in space (O't(r.) = O't), so that the parameter t: is constant on the computational domain D := 'R x 81 where R. C JR:l is a region with sufficiently smooth boundary, for example, of class Cl,l (Grisvard [22, p, 5]), and 51 denotes the unit sphere. Further, we suppose throughout this chapter that C ::; l. Moreover, in the following we restrict our attention without loss of generality to problerns with vacuurn boundary conditions (so g(z:,l) = 0 in (L4)). Problems with inhomogeneous boundary conditions can easily be transformed to problems with homogeneous boundary conditions and different right-hand sides (Oden and Carey [45, p, 27]). As in the one-dimensional case) a scaling transformation is applied to the transport operator prior to the Least-Squares discretization to ensure accuraey of the discrete solution in diffusive regions. In the three-dimensional case) the scaling transformation and its inverse are given by S := P + c(l -P); s-1 := P P). [ (3.1) They have the same form as in the one-dimensional case with the only difference that r = c and that the L2-orthogonal projection P onto the space of fUnctions that are independent of direction vector D. is now defmed by N = j PAGE 55 inner product (u, v)v := Jjl 1 1. 1 ;:SU V'u -'Sn V'v + -(IP)u -(IP)v v [--[ [ "R s< (3.4) + Pu Pv d(ldr, its associated norm llullv = V(u,u)v = V'ull' P)ull" + IIPuli') 112 (3.5) and the space V := { 1; E C00(D); v(r., Q) = 0 for r E 8R, and Q rr(r) < 0}, (3.6) where the closure is with respect to the norm llllv, so that it is a Hilbert space. As mentioned in the introduction) the Least-Squares variational formulation of (3.3) is given by (1.19). Our first goal is to show that the bilinear form a(,), defined in (1.19) is continuous and V-elliptic with constants independent of pararneter E and a. From these results will follow not only the well posedness of problem (1.19), as outlined in the introduction, but also the accuracy of the Least-Squares Finite-Element discretization applied to it in the diflusion limit. 3.1 Continuity and V -ellipticity As in Chapter 2, we conclude from the Cauchy-Schwarz inequality that the bilinear form a(,) is continuous, since ia(u, v)l = I(.Cll, .Cv)l <::; II.Cuiiii.CvllWe now use the discrete Holder inequality to get II.Cull <::; V'ull P)ull + IIPull <::; V3 (II V'ull' + II P)ull' + 11Pull2 ) 112 v'3 llullv, since a: ::; 1 by a,.<;sumption. Thus, for any u, v E V, la(u,v)l <::; C, llullv llvllv, (3.7) with C, = 3. For the more difficult part of the proof of the V-ellipticity of a(,), we proceed in the same way as in the one-dimensional case. We first scale (3.3) in addition from the right by S to get .cs$ = !sos. v;j; +(IP)$+ "'P$ = q, 47

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where -;j; := s-1'1/J. Define the new space V and associated norm by where v := s-1v, 1 Q = -SflS = (1s) (Pfl + flP) + Efl!. -[ Shortly, we will prove that the bilinear form a(u, v) := j j J:Su csv df.ldr 11 il,:,; E v n S' (3.8) (3.9) is V-elliptic. The V-ellipticity of a(.,) in (1.19) will then follow in the same way as in Corollary 2.12. Before we do this, we first establish the following lemmas, which are generalizations of Lemma 2.2, Lemma 2.4, Lemma 2.9, and Lermna 2 .10. Le1nrna 3.1 For all u,v E V, VE V and::; 1, we have (i) (Pu, v) = (u, Pv); and ((IP)u, v) = (u, (I-P)v); p2 = P; and (IP)2 =(IP). Thus P and (I P) are orthogonal projections. (ii) (Pu, v) = (Pu, Pv); and ((IP)u, v) = ((IP)u, (IP)v); (iii) llvll II ; (iv) IIPvll-ell( IP)vll < llvll; (v) (fl 'Vv, v) 2: 0; (vi) (9.. 'Vv, v) 2: o. Proof. The proofs of (i), (ii), (iii), (iv) follow analogously to that in Lemma 2.2 and Lemma 2.9. To prove (v), we apply the fundamental Green's formula (Ciarlet [16, p.34]) to get (fl v) = j j fl 'Vv v df.ldr n S' = -j j v fl \7 v dfldr + j ju' fl 'l dllds. n sl an Sl We therefore have (fl 'Vv, v) = j j v2 fl 'l df.lds ans1 48

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Splitting the boundary an X 51 into the parts r+ := { (r:, m E an X 51 ; g. rr(r:) 2: 0} and r-:= { (r:,!}) E an x 51 ; il 11(1:) < 0}, the boundary integral becomes j / v2l.l = j j v2il 'l dllds + /.I v2il 'l dOds an 51 r+ r= j j v2il 'l d(lds 2: 0, r+ since v(r:,!}) = 0 for (r:,!}) E r-and v E V. Thus, altogether we obtain (il v) 2:0 To prove (vi), we observe from (i) that Sis with respect to(,). Con sequently, we have (Q 'Vv,v) = Gsns-vv,v) = vsv,Sv) = 2: o, where the last inequality follows from (v). D Lemma 3.2 (PoincareFriedrichs Inequality) Suppose E ::; 1 and let n c IR3 be a bounded domain. Then, for any v E v' we have llvll ::; diarn(n) 1151 'Vvll ::; diam(R) IIQ 'Vvll-(3.10) Proof. For L, Lk E 1? let denote the line segment between r..i and r.k. Let arbitrary r.. E R and Q E 51 be given. We define t, min{t E lR: [r:,r:+tQJ C n} t2 max{tElR: [r:,r:+tl.l]cn} r1 r. + ftil; r2 ::::: r.. + Then it is easy to see that .Q_ rr(r:1 ) < 0. Taking into account the boundary conditions for v E V, we therefore have that v(.r:1 ) = 0, hence, av jc v(r n) = ds = n _,_ Os ds, where ds denotes the arc-length differential along the line {r: +til, t E lR}. Therefore, we conclude from HOlder's inequality that ( ) 1/2 lv(r:,il)l :0: J 151 'Vvl ds :0:7151 'Vvl ds :0: diam(n)112 J lil 'Vvl2 ds 49

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Applying Fubini's theorem, it follows that r., J J lv(r, ml' diJdr :S diam(R) j j jIll Vvl2 diJdrds n S' !:.1 R Sl :S diam(R-)2 j jIll Vvl2 dndr. R 51 From the relation v = SV and (iii) of Lemrna 3.1, we thus have II vii :S diam(R) I Ill Vvll = diam(R) I Ill \7 Siill :S diam(R) II Vvll = diam(R) IIQ Viill, which proves the lemma. 0 Lemma 3.3 Suppose 0 :Sf :S 1. Then, for any o E [0, !], there exists b?. 0 such that where s := [o1i2 -s(l 15)112 ] 2 In particular, for .5 < 0.875, we can choose b = 0. ]' Proof. The only difference to the proof of Lemma 2.10 is that now s = [o1i2 -c(l-8)112 instead of s = [8112 -svf:l(1-8)112 ]2 Therefore, when 8 > 0.875, we use the assumption :S 1 to get s ?. 1-2Jd(1-d) =: (3. Everything else is analogous to the proof of Lemma 2.10. D We are now in a position to state the central result of this section. Theorem 3.4 (V-ellipticity of a(-,)) Let a(-,) and llllv be given as in (3.9) and (3.8). Suppose that 0 :S a :S 1, 0 :S f :S 1 and that the diameter diam(R) of the domain1 R is 1. Then there exists a constant C, > 0 such that, for all v E V, a(v, v) = 119. + aPv +(IP)vll' (3.ll) where C'e = 0.012, which is independent of E and 0'. Proof. ln the proof of Theorem 2.11, we replace Q Q_ Vii and for Panda(-,) use the definitions of this chapter. Then the proof of Theorem 3.4 follows exactly as the proof of Theorem 2.11, except that the PoincareFriedrichs inequality (3.10) of Lemma 3.2 and (iv) of Lernma 3.1 are now used to get 1 This can be established by a simple transformation of the space coordinates r_. 50

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Therefore, s in (2.47) is replaced by s = [v'b-cVf=8]' and Lemma 3.3 is applied to bound the functions Gr and G,. D From the V-ellipticity of the bilinear form a(,), the V-ellipticity of the bilinear form aC, )follows immediately as in Chapter 2. 'We summarize this result in the following corollary. Corollary 3.5 ( V-ellipticity of a(-,)) Let a(-,) and llllv be given as defined in (1.19) and (3.5). Suppose that 0 :S <> :S 1, 0 :S E :S 1 and that diam(R) = 1. Then there exists a constant C, > 0 such that, for al111 E V, (3.12) where Ce = 0.012, which is independent of a and s. D 3.2 Spherical Harmonics Since a truncated expansion into spherical harmonics (.PN-a.pproximation) is used throughout this chapter for the the disretization in angle, we introduce here the spherical harmonics and summarize important properties that are needed for the error bounds. Recall that the associated Legendre polynomials are defined for l 2 0 and m = 0, ... I by (Margenau and Murphy [39, p. 106]) (3.13) where Pr(f.l) is the (unnormalized) Legendre polynomial of degree I. By the formula of Rodrigues (Arfken [3, p. 554]) for the Legendre Polynomials given by 1 d1 ( 2 )' P,(p) = 211! dp1 I' 1 this definition becomes (3.14) Expression (3.14) can be used to extend the definition of P,m(p) to negative integer values of m. It follows that P1m(f.l) and P1-'"(p) are related by -m( ) ( )'"(Im)! pm( ) P1 I' = -1 (/ )I 1 1'+m. (3.15) The associated Legendre Polynomials satisfy the following recurrence relations ( Ar fken [3, p. 560]): 1 21 + 1 [(I+ m)P{".1(p) +(1-m+ 1)Pf+1(p)], (3.16) 1 (pm+l( ) pm+l( )] 21 + 1 1+1 I' -1-1 I' (3.17) 1 21 + 1 [(I+ m)(l + m-1)P("_j1(p) -(I-m + 1)(1m + 2)P,'_;:j1(p)]. (3.18) 51

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These recurrence relations) although derived in (Arfken [3]) only for positive integers m) remain valid for negative values of m. This can be easily checked by substituting in the relation (3.15) into the left and right parts of the recurrence relations. Further, the associated Legendre polynomials satisfy the orthogonality relation 1 / P,m(l') P;:' (fl.) dp = -1 2 (i+m)'0 + 1 (1-m)' lk. (3.19) Based on the associated Legendre Polynomials the spherical harmonics are de fined by (Arfken [3, p. 571]) where (21 + 1)(1m)' (I+ m)1 (3.20) Here, 0 denotes the polar angle with .respect to the z-axis, while r.p denotes the azirnuthal angle about the z-axis is. The spherical harmonics form an orthonormal basis of L2(S1): In particular, z, // Y{"(O,"'*(O,
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where Ct:[,rn /l,m (l+m+2)(1+m+l) 4(21 + 1 )(21 + 3) (1-m)(l+m) (211)(21 + 1)' f3z,m 1]/,m (1m)(l-m-1) 4(2/-1)(21 + 1) (/+m+l)(l-m+l) (2/ + 3)(2/ + 1) = Cl,m Using for the first term recurrence relation (3.17) and for the second term relation (3.18), after simple but tedious calculations we obtain the first recurrenee relation. The second recurrence relation follows in a way similar to the first. For the third relation, recurrence relation (3.16) is used. 0 Since the spherical harmonics form an orthonormal basis of L2(S1 ), every v E H'(R.) x L2(S1 ) has an expansion of the form oo I J v(z:,Q) =I: I: l,m(z:lYnQ), with l,m(Z:) = v(z:Jl)Yj"''(Q)dO. (3.25) 1=0 m=-1 51 For any v E H'(R.) x H2(S1 ), we define N-1 I fiNv(z:,Q) := I: I: ,m(z:)Y1 m(ll), l=O m=-l with l,m(z:) = ./ v(z:,Q)Y,m'(ll)dO 5' (3.26) as the truncated expansion of v into spherical harmonics. To bound the error of the truncated expansion, in the following lemma we use the fact that the spherical harmonics are the eigenfunctions of the Laplacian operator on the unit sphere, so (3.27) = -1(1+ 1)Y;"'(Q) for I:?: 0 and m = -I, -I+ 1, ... 0, ... I. Lemma 3.7 (Truncated expansion into spherical harmonics) Let (3 be any multi-index and recall that DDv := Suppose that v(z:,ll) E HIPI(R.) x H2(S1 ) and, for N:?: 2, let fiN be defined as in (3.26). Then IID8,mll < l) IID-nDPvll for I:?: 0; -1 m (3.28) c N IID-nvll, (3.29) 53

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with C independent of v and N. Proof. By the definition of
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with functions that have a truncated expansion into spherical harmonics with respect to direction angle D. and are piecewise polynomials of degree k on a triangulation of the region 1?... into tetrahedrons. This choice of finite-dimensional spaces corresponds to a discretization by a spectral method in direction angle ?. and a Finite-Element discretization in space. In transport theory, the spectral discretization in angle using spherical harmonics as basis functions is also called a PN-approximation. Let Th be a triangulation ofR. into tetrahedrons T of maximum diarneter h. For any v(c:, mE H"+1(R) X 2(51), let lhv denote the interpolant of v by piecewise polynomials of degree ron the triangulation Th Then, similar to (2.27), it can be shown (Ciarlet [16]) that, for 0 :<:; rn :<:; 1, (3.30) where llllm 0 denotes the standard norm of Hm(n) xL2 (51 ) and I lr+!,O denotes the standard semi-norm 'of Hr+1(R) x 2(51 ) (see Section 1.4). In order to combine Cea's Lemma with (2.29) and (3.30) to obtain a discretization error bound, we need the following lemmas. Lemma 3.8 (Bound for commutator [IIN.C-.CITN]) Suppose N 2: 2 and let the operator b.n be defined as in (3.27). Let v E H1(R) x H2(51). Then there exists C > 0 independent of a and E such that Proof. By expansion (3.25), it easily follows that IINPv = PIINv and IINP!J. \lv P!J. \1 !I NV. Therefore, (3.32) Now using the recurrence relation (3.24) in Lemma 3.6, we get = !IN [.;:;-. iJ
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so that N 8 '\' (f3 ym+l 6 ym-1) L 8x N,m N-1 t N,-m N-1 m=-N N-1 I: 8
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and We now continue by bounding the sums in the following way: and N .L "(N,m -N N-l N 2 "'. VN' 2 :S 2N -1 + 2N -1 L-m m=l N 2 < + (N -l)N = N -2N -1 2N-1 N-1 N-1 .L 'IN-l,m = .L N m=-N+l m=-N+l (N + m)(Nm) < _L 'YN,m :S N, (2N + 1)(2N-1) -m=-N so that :z liN vii which proves the lemma. D Lemma 3.9 3/[RT II A ov II < --un-1. N oz Let V and llllv be given a.s defined in (3.8). Then for all v E V n (H1(R.) x L2(S1)): with C independent of and a. Proof. By definition, it follows that ( 2 ')1/2 lliillv :S [(1r) {IIPQ 'Vvll + 110. P'Viill} + c 110. iilll +I Iii II Notice that since lllxl :S 1, lily I :S 1, and Ill, I :S 1. Similarly we have and IIPQ 'Viii I :S 110. 'Villi :S Vslliill,,o 57 (3 34)

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From these bounds, (3.34) follows immediately with C = ( [3V3] 2 + 1 )'12 = y'28. D Now we are in a position to establish the following error bound. Theorem 3.10 (Finite Element in space, PN in angle ) Let Th be a triangulation of R. into tetrahedrons of maximal diameter h. Suppose 0 ::=; a :::; 1, 0 :S E :S J3 and diam(R) :S 1. Let V be given as defined in (3.6) and let V" be defined by where IP r (Th) denotes the space of piecewise polynomials of degree _:::; ron the triangulation Th. Let 1/J E V n (W+l(R) x H2(S1 )) be the solution of (1.19) with q, E U(R) x H2(S1 ) and let 1/Jh E V" be the solution of (1.19) restricted to yh Further assume that 1/J has the diffusion expansion 1/J(r_, ll) = cf(r_) + cn(r_, ll). Then .;c; c, I I I I J 111/J -1/Jhllv :S C, N (I D.nq, I+ D.n1/J 1,0 +if G'2 h' (lcflc+l,O + lc/>nlc+l,O), with cl and c2 independent of Ct' and[. Proof. By Cea's Lemma, we have 111/J -1/J"IIv :S ffj 111/J-liNflh1/JIIv :0: ffj (II= JihiT-;-1/J. Therefore, llliN1/J-n,rrN
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where the last inequality follows from (3.34) in Lemma 3.9. We now use (3.30) and the diffUsion expansion of 1/; to get (3 38) Remark 3.11 (Nondiffusive regimes) In Theorem 3.10, it is assumed that the analytical solution has a diffusion expansion in order to get an error bound in (3.38) with a constant that is independent of parameter c. For regimes where the diffusion expansion is not valid, is of rnoderate size, so that there is no need for an error bound that is independent of s. Therefore, in this ca..'Se, (3.38) can simply be bounded by :S Chr ( 1 + j,Pjr+l,O > so that the overall bound becomes 0 59

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CHAPTER 4 MULTIGRID SOLVER AND NUMERICAL RESULTS According to the theory derived in our earlier chapters, the Least-Squares approach yields accurate discrete solutions, even for diffusive regimes. In this chapter we confirm this result by numerical tests and demonstrate that the resulting discrete system can be solved very efficiently by a full multigrid solver. The following tests are restricted to the one-dimensional transport problem (1.6). For the discretization in angle, a PH-approximation is used, which is a spectral method using the first N Legendre polynomials as basis functions. For the discretization in space, we employ a Finite-Element discretization with linear or quadratic basis functions. To be more precise, we recall that the analytical solution has the rnoment expansion 00 1/J(z, I')= L
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Recall that, for a function v E VN we can use a Gauss quadrature formula with N support abscissas {,ttl, ... f-LN} and weights {w1, ... w N} to write 1 N Pv = j v(z,J1)dJ1 = Lw; v(z,J1;), -1 J=l ( 4.3) since this quadrature with N support abscissas is exact for polynomials of degree ::; 2N -1 (Stoer and Bulirsch [49, p. 153]). In the SN-cliscretization, the flux is only cornputed for the discrete set of angles {t-tl, ... t-tN }, so that the unknowns are given by the vector By collocating at the Gauss points and approximating the operator P by the sum in ( 4.3), the following SN-flux equations for 1 can be derived from tbe transport equation (1.6): where q .-( q(z,J1t) ) ( 1 ) q(z,l'N) l.= 1 M := diag(l't, ... J1N ), R := lw T Further, for v E V N we note that the scaling transformation S notation becomes ( 4.4) P + r(I-P) in this with IN denoting the N x N identity matrix. Therefore, the scaled SN-ftux equations are given by (4.5) with 'L := SN'l_ The boundary conditions for (4.4) and (4.5) are given by and ( 4.6) respectively, where I!::!.. denotes the !;f x f identity matrix. 61

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4.1.2 Moment Equations In order to derive the moment equations from the :flux equations, we note that, for 1/J E VN, N-l 1/! = L
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Proof. N 1 (i): ""T l = I: w; = I 1 dfJ = 1. Therefore, R2 = lw T lw T = l("" Tl)"" T = 1w T = R. j::::::l -1 (ii): RTO=wlT'l=wwT=D1wT=OR. (iii): (iv ): ( v ): N 1 (Tm'T)1 .=I: Pi-1(Jlk)w,p;-1(Jlk) Pi-1(JJ)P;-1(JJ)dJJ = 6;; ,J k::::::t -1 Therefore, TOTT = I, so TT nonsingular =;. 3C such that TT C = I =;. T(lTT C = TD =;. C =TO=;. TTT(l =I. N N 1 (7'"")1 =I: Pi-1(JJ;)w; =I: Pi-1(/;j)w;Po(JJ;) = [ Pi-1(JJ)po(JJ)dJJ = 6;1. j::::l j:;;;;l -1 The unnormalized Legendre polynomials P,(JJ) satisfy the recursion (Arfken [3, p. 540]) (4.10) Since the normalized Legendre polynomials are given by pz(tt) := .J2l+TP,(JJ), from (4.10) we have fJPl(JJ) = bz-1P1-1(11) + blPI+1(J1) with bz-1 := .;4/,_1 Using (4.11), we then have Therefore1 k::::::l 1 1 (4.11) = j Pi-r(JJ) P;-AJJ) dr + b;-1 / Pi-1(!)P;(JJ) dfJ -1 -1 (vi): T(JRTT = (1'rll) ("" TTT) = T"" ""TTT = (1, 0, ... 0) T (1, 0, ... 0), where we used (iv) in the last step. D Multiplying the unsealed flux operator iL by TT from the right and by TO from 63

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the left and using Lemma 4.1 gives TOfiTT = TOMTT :z + cr, (nliNTT-TORTT) + craTORTT 0 : ll [ : 0 : l 0 0 +"a 0 0 ( 4.12) era l 8 "' j =B-+ -. IM, az "' where IM is the unsealed moment operator. Therefore, multiplying ( 4.5) by TO fi-om the left and using ( 4.9) results in the following scaled moment equations: Trl1L1/:_ = TrlSNTTnlfiTT '!!_ = (TOR:fT + rT(lTT-rTOR7T) .IM '!!_ (4.13) Again using relation ( 4.9), it follows from ( 4.6) that the boundary conditions for the moment equations are given by (4.14) We conclude that the SN-flux equations and the PN-moment equations are equivalent serni-discrete forms of the transport equation. The difference between these two sets of equations is that the non-derivative part in the flux equations is fully coupled, while the derivative part is decoupled. For the moment equations, the reverse is true. 4.1.3 Least-Squares Discretization of the Flux and Moment Equations After deriving the SN-ftux and moment equations we return to the discrete PH problem (4.2). Using a Gauss quadrature formula with weights {w1, .. ,wN} and points 64

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{Jt1 1 1 J.lN} to approximate the integration over angle J.l results in Zr N ./ f;; Wj [,h,(z, !')] (Jlj) [bk,i(z, l')](l'j) dz ,, ( 4.15) z, N = ./ L_wjq,(z, l'j) [bk,l(Z,!L)](!lj) dz. Zl J .::::;1 We note that on the left-hand side1 the approximation of the integration over angle p by the Gauss quadrature formula is exact as long as l < N1: since then .Cbk,l is a polynomial in I' of degree I+ I, while /;h is a polynomial in I" of degree N, then the product is a polynomial in I' of degree S 2N I, for which the Gauss quadrature formula with N support abscissas is exact. Therefore1 only in the equations for { bk ,N -11 k = 01 1 m} must we introduce an error on the left-hand side by approximating the integration over angle. On the other hand, the same argument shows that the right-hand side is represented exactly by the Gauss quadrature formula as long as q, (z, I') haB an expansion into the first N -2 Legendre polynomials. With the notation introduced in Section 4.1.1 we have ( [/;;,(z,p)] (!'r) ) [/Jh(z, !')] (J"N) and where t41 denotes the (I+ I )-nth column of the matrix TT defined in Section 4.1.2. Denoting by(, )JRN the standard Euclidean inner product of IRN, (4.15) then becomes ,, ./ \rliL1j;_h,ILryk(z)t4r)JRN dz (4.16) z, = .// f:lq ILryk(z)tf+r) dz \ -s JRN for all k E {0, I, ... m} and IE {0, I, ... N-1}. Since the columns of TT span IRN, then we can substitute {:t.J, ... 1 t"J,;.} by the canonical basis {1 ... fiN} of IRN and we recognize that ( 4.16) is a Least-Squares discretization of the 8N-flux. equations using the discrete space (4.17) This is the space of N-vector functions whose components are piecewise linear (for r = 1) or piecewise quadratic (for r = 2) polynomials on the partition Th of the slab. 65

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Using (4.9) and (iii) of Lemma 4.1, we can rewrite (4.16) as ' J (neTT _h= '. ,q\f(z)=I:>k,l1)k(z),fori=O, ... ,N-1 q\)\,_I(z) k=O (4.19) All computations in the following sections are based either on discrete problem (4.16) or (4.18). 4.2 Properties of the Least-Squares Discretization In this section 1 we use the results of numerical experiments to observe properties of the Least-Squares discretization. The results plotted in Figure 4.1 and Figure 4.2 demonstrate the accuracy of the Least-Squares discretization in combination with the scaling transformation for diffusive regimes. The test problem we chose here is the same one used by Larsen et al. in [32]. The exact solution of the corresponding diffusion equation is q\(z) = -3/2z2 + 15z, which is plotted in solid in Figure 4.1 and Figure 4.2. The scalar flux .Po := Pl/Jh of the solution ljJ, of the Least-Squares discretization of the scaled transport equation using piecewise linear elements in space is shown by the crosses. For the problem in Figure 4.1, where the absorption cross section is zero, we used r = 1/a-; = c2 as the scaling parameter, which gives a higher accuracy than the scaling with r =c. An explanation of this result is given in the analysis presented in (Manteufl"el and Ressel [38]). For the test problem in Figure 4.2, where
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with piecewise linear basis elements in space is a straight line connecting the values at the boundary in diffusive regimes. Moreover, the asymptotic analysis in Theorem 2.1 asserts that the Least-Squares discretization of the unsealed transport equation using piecewise polynomials of degree 2: 2 in space has the correct diffusion limit. This too is supported by the observed maximum errors for a Least-Squares discretization of the unsealed transport equation with piecewise quadratic elements in space, which we list in Table 4.1. However, using the scaling transformation in combination with the Least-Squares discretization with quadratic elements in space achieves dramatically better accuracy in the discrete solution. For piecewise linear elements in space, the error bound in Theorem 2.14 indicates an 0( h2 ) behavior of the Least-Squares discretization error for a sufficient srnooth solutions. To analyze the order of the Least-Squares discretization numerically, we used a problem with smooth exact solution sin(1rz). We then computed the discrete L2-error of the Least-Squares discretization with linear elements in space for a sequence of grids that were created from the coarsest grid by halving the mesh size from one to another grid. Table 4.2 depicts the ratio of these errors for each two consecutive grids. The value of approximately 4 of this quotient confirms numerically an O(h2 ) behavior of the discretization error for linear elements. The solution of the transport equation is physically a density distribution and should therefore always be positive. The Least-Squares discretization has the drawback that it does not in general guarantee a positive solution. This is shown by the example in Figure 4.3, where the exact solution of the corresponding diffusion equation is again plotted as a solid line and the discrete Least-Squares solution is depicted by the crosses. Of course, this boundary layer can be resolved by refinement of the mesh, as shown in Figure 4.4. IIowever, in the region [2, 10], the solution is nearly constant, so that a refinement makes sense only in the region around the boundary layer. Therefore, the aim is to use adaptive refinement, which can be combined very naturally with a full multigrid solver (McCormick [42]). One easy criterion for determining the area of further refinement would then be to check where the solution is negative. Of course1 this has to be combined with more sophisticated criteria that compare the solution of consecutive grids, for example. Besides having the correct diffusion limit, a discretization for transport problems must satisfy the extra condition to resolve, with a suitable fine spatial mesh, interior bound ary layers between media with different material cross sections. To test numerically if the Least-Squares discretization meets these extra conditions, we used the test problem from (Larsen and Morel [33]), which is given in Figure 4.5. The solid solution plotted in Fig ure 4.5 is computed by a Least-Squares discretization using 50 cells in both [0, 1] and [1, 11]. This solution approximates the exact solution plotted in (Larsen and Morel [33]) fairly well. We see further that the boundary layer is not resolved fully when the mesh spacing for the Least-Squares discretization is too coarse (crosses in Figure 4.5). In addition, the Least Squares solution itself indicates an euor by becoming negative. Again adaptive refinement would be an appropriate remedy. 67

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Scalar Flux 40 35 30 25 20 15 10 0 2 3 4 5 6 7 8 9 10 Test problem: { 81/! } Paz+ 100(1-P)

O V;(lO,p) = 0 for p
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35 30 25 20 15 10 Scalar Flux 2 3 4 5 6 7 Test problem: { 1' + lOO,P-99.99Ptp = 0 01(1 + ) } 1/;(0,1') = 0 for I'> 0 ,P(lO, I')= 0 for I'< 0 (c = 0.01, "= 1.0) 8 9 Solution of corresponding diffusion equation: (x) = -3/2 x2 + 15x. 1 I I l 10 Figure 4.2: Scalar Flux of exact (solid) and Least-Squares solution with scaling transformation (crosses) and without (asteriks). 69

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Table 4.1: Comparison of maximum error for scaled and unsealed Lea.c;t-Squares discretization with piecewise quadratic elements in space. Ci = 0.0 = 1.0 1/ h sca.led unsealed scaled unsealed 4 1.8-4 5.2-2 1.2-4 3.2-2 8 1.2-5 1.2-lo-2 7.7-6 7.7-3 16 7.7-7 3.2-3 4.8-7 1.9-3 32 4.8-8 7.8-4 3.0-8 4.6-4 64 3.0-9 1.8.10-4 1.8-9 1.1-4 128 1.8-10 3.8-lo-5 1.1-10 2.2-5 256 1.3-11 6.3-6 7.9-12 3.8-6 Test problem: { [Pf, + (1,p) = q for (z,l') E [0, 1] x [-1, 1] } = 0 for fl.> 0 = 0 for p < 0 where at= 1000.0) a a=...) q := p7rcos(1rz) +a a sin(1rz)) a, Exact Solution: t/>(z, p) = sin(1rz), Number of Moments: N = 2. 70

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Table 4 2 Order of Least Squares discretization for linear elements in space -, 1 h 4 8 16 32 64 128 256 512 1024 2048 4096 8192 E = 1.0 E = 0.001 0! = 0.0 0! = 1.0 0! = 0.0 0! = 1.0 /lehll2 llehll2 llehll2 I' llehll2 8.5 2 2.6 2 1.5 2 1.2 2 2.1-2 4.0 6. no-3 3.9 3.8-3 3.9 2.9-3 5.4-3 3.9 1. no-3 3.9 9.7-4 3.9 7.5-4 1.3-3 4.1 4.2-4 4.0 2.4-4 4.0 1.8-lo-4 3.4-4 3.8 1.0-4 4.2 6.1-5 3.9 4.7-5 8.5-5 4.0 2.6 10-5 3.8 1.5-5 4.0 1.2-5 I 2.1-5 4.0 6.6-6 3.9 3.8-6 3.9 2.9-6 5.3-6 3.9 1.6-6 4.1 9.5-7 4.0 7.3-7 1.3-6 4.0 4.1-7 3.9 2.4-7 3.9 1.8-7 3.3-7 3.9 1.0-7 4.1 6.0-8 4.0 4.6-8 8.2-8 4.0 2.5-8 4.0 1.4-10-8 4.2 1.).10-8 2,0-8 4.1 6.5-9 3.8 3.5-9 4.0 2.8-9 Test problem: { [Pt, + O"t (IP) + O"aP],P(z,p) 1/;(0,p) 1/;(l,p) = q for (z, p) E [0, 1] x [-1, 1] } = 0 for p > 0 = 0 for p < 0 where Ut = 1\ eTa= q := jl/ifcos(7rz) +eTa sin(1rz)\ o, Exact Solution: 1/J(z,p) = sin(rrz), Number of Moments: N = 4. 71 4.1 3.9 4.1 3.8 3.9 4.1 3.9 4.0 3.9 4.2 3.9 I

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Scalar Flux 0.8 0.6 0.4 0.2 0 0 2 3 4 5 6 7 8 9 10 Test problem: { f.l + 100;&-99.9P,P = 0.0 } ;&(O,J.l)=1 for J.L>O (b(lO,f.l) = 0 for J.l < 0 (c = 0.01, "= 10.0) Solution of corresponding diffusion equation: Mesh size: h = \s0 1 Moments: N = 4. }t'igure 4.3: Example of violation of the positivity property by the Least-Squares discretization 72

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1, 0.9 0.8' 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 Scalar Flux 3 4 5 6 7 Test problem: { J1 + 100<,0-99.9P,P = 0.0 } <,0(0, p) = 1 for I'> 0 <,0(10,p)=0 for p
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Scalar Flux 0.15 0.1 0.05 0 X ... x. -0.05 -0.10 2 3 4 5 6 7 8 9 10 Test problem: { 81/; } !' oz + O 1/;(ll,f.l)=O for f.l < 0
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4.3 Multigrid Solver In this section we describe the multigrid solvers, that were developed for solving the problems resulting from a Least-Squares discretization of SN-flux (4.16) and moment (4.18) equations with piecewise linear elements in space. We refer the reader who is not familiar with multigrid methods to (Briggs [6]) for an introduction and to (Hack busch [24]) and (McCormick [41]) for more advanced topics. Essential for the efficiency of a multigrid solver is the proper choice of its cornpo nents, mainly the intergrid transfer operators, coarse grid problems, and relaxation schernes. The choice of the -first two components is naturally given by the Least-Squares variational formulation: the sequence of discrete spaces vl. c v2 c ... c Vi = yh determines the coarse grid problems since they are just the restriction of the variational problem to these discrete subspaces; the prolongation operator: which is a mapping from a coarse grid to the next finer grid in the grid sequence: is formed directly by composing the isomorphisms between the discrete spaces and their corresponding coordinate spaces with the injection mapping between Vk-1 and Vk (Bramble [5]), (McCormick [43]); and the restriction operators, which are mappings from a finer grid to the next coarser grid, are just the adjoints of the prolongation operators. Therefore, the only rnultigrid components that need to be chosen here are the sequence of discrete spaces and the relaxation. No relaxation scheme is currently in use for transport problems that smooths the error in angle and in space simultaneously. 'I.'hus, instaed of devising a :rnultigrid scheme that coarsens simultaneously in space and in angle, we consider first applying the multilevel-in angle technique of (Morel and Manteuffel [44]), which is based on a shifted source relaxation scheme. After reducing the degrees of freedom in angle, a multigrid method in space is used to solve the remaining discrete problem. Thus, here we consider only the development of a rnultigrid solver in space. For the discrete subspaces, we use the Finite-Element spaces with linear basis elements on increasingly finer partitions (halving the cells) of the slab. 4.3.1 SN Flux Equations The stencil that results from a Least-Squares discretization of the SN flux equations (4.16) with these Finite-Element spaces is given in Appendix A and shows full coupling in angle. This suggests the use of a line relaxation in angle, which updates all angles for a given spatial point simultaneously. The matrix that must be inverted for each spatial point for this scheme is of the form (see Appendix A) The first part is diagonal, and the second has the rank 2 factorization ) Thus, Ai can be cheaply inverted by the Sherman-Morrison formula (Golub and Van Loan [20, p. 51]). Our computational tests showed essentially no differences in the error reduction and smoothing properties of this line relaxation seheme for various different orderings of the spatial points. To save computational, we thus use this line relaxation scheme in a red-black fashion, since then the residual after one relaxation sweep is 7,ero at the black points and 75

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not need not to be computed for the restriction to the next coarser grid. This scheme is also more amenable to advanced computer aechitecture efficiency. The convergence factors for this rnultigrid algorithm 1 listed in Table 4.3, are computed in the following way. A problem with zero source term and and whose exact solution equal is zero is used in combination with a randomly generated initial iterate. Then 30 rnulti grid cycles are performed and the convergence rate is computed from the geometric average of the per-cycle reduction factors of the last 20 cycles. We thus reduce the influence of the initial iterate on convergence and observe what tends to be the worst-case factors. Here we study the (1, 1)-V-cycle, which uses one relaxation before and one after coarse grid correc tion. Observed factors for crt S 106 are on the order of 0.1 for all values of the absorption coefficient cx. Factors for (2,1)V-cycles are also included. Such factors are sufficient to get a solution with an error on the order of the discretization error by one full multigrid cycle, as demonstrated by the results in Table 4.4. The additional V cycle on the finest level 10, performed subsequent to the full multigrid cycle, is reducing the error only by a small amount. Thus, we can conclude, that the error after the full multigrid cycle is completed is already on the order of the discretization error. 4.3.2 Moment Equations The stencil for the Least-Squares discretization of the moment equation ( 4.18) is given in Appendix B. In the interior of the computational domain, it is a 15-point stencil that connects the neighboring spatial points and the two higher and two lower mmnents. At the spatial boundary, however, the stencil couples all mornents. Therefore, we use a line morr.tent relaxation, that updates all rnoments simultaneously for a given spatial point all mornents simultaneously. Since the efficiency of the smoothing again is observed to be independent from the relaxation ordering, as in the SN flux case, we use a red-black ordering of the lines. The convergence factors for this multigrid algorithm, are listed in Table 4.5. For very large values of at, this multigrid solver is more stable with regard to roundoff errors than the multigrid solver for the SN flux equations. Even for values of O't 2:: 106 we get (1, 1)V-cycle convergence factors of order 0.1. Again, these convergence factors are sufficient to get a solution with an error on the order of the discretization error by one single full multigrid cycle, as demonstrated in Table 4.6. 1This algorithm was implemented in C++ and a special array class was designed for this purpose. 76

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Table 4.3: Multigrid convergence factors for solving the flux equations. ( 1,1 )V -cycle ITt = 1.0 "= 0.5 "= 0.25 "= 0.1 "= 0.0 10 0.088 0.085 0.087 0.118 0.169 101 0.082 0.083 0.083 0.110 0.136 102 0.052 0.052 0.053 0.106 o.1:Jo 103 0.088 0.091 0.088 0.105 0.130 101 0.091 0.091 0.091 0.105 0.130 105 0.092 0.092 0.092 0.105 0.130 106 0.090 0.092 0.092 0.102 0.133 (2,1)-V-cycle ITt 1.0 a-0.5 "-0.25 0.1 (Y-0.0 10 0.053 0.050 0.053 0.105 0.155 101 0.047 0.047 0.047 0.082 0.104 102 0.019 0.024 0.024 0.077 0.097 103 0.020 0.021 0.021 0.076 0.096 104 0.020 0.022 0.022 0.076 0.096 105 0.020 0.011 0.023 0.076 0.096 10' 0.019 0.023 0.018 0.077 0.099 Test problem: { [l'iz +ITt (I-P) + rraP] 1/J(z,JL) 1/J(O, JL) 1{;(1,;<) = 0 for (z,JL) E [0, 1] x [-1, 1] } = 0 for Ji > 0 = 0 for Ji < 0 where era::::::: Exact Solution: 1/J(z,JL) = 0. Initial Iterate: randomly generated grid function. Mesh size: h::::::: Number of Moments: N = 8. 77

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Table 4.4: Full Multigrid (1,1)-V-Cycle convergence factors for solving the SwRux equa tions. ITt 1.0 Level t Ci-1.0 a-0.5 a-0.0 lhll2 k llehllz k lieh 112 k 0 2 1.06. 10 ., 2.96. 10 3.65"' 1 4 3.90. 10-2 2.7 8.85 to-3 3.0 1.79 10-1 2.1 2 8 1.04. 10-2 3.7 2.88 to-s 3.0 6.32. L0-2 2.8 3 16 2.68. to-s 3.9 7.47. 10-4 3.8 I 1.92. 10-2 3.3 4 32 6.80. 1o-" 3.9 1.88. 10-4 3.9 5.36 to-s 3.6 5 64 1.71. 10-4 3.9 4.72. 10-5 4.0 1.43. 10-3 3.7 6 128 4.29. to-s 4.0 1.18 to-' 4.0 3.72. w-" 3.8 7 256 1.07. 10-5 4.0 2.95. to-s 4.0 9.52. to-5 3.9 8 512 2.69 w-s 4.0 7.38. 10-7 4.0 2.41 to-5 3.9 9 1024 6.72. w-7 4.0 1.84. w-7 4.0 6.08 lo-s 4.0 10 2048 1.68. 10-7 4.0 4.60 lo-s 4.0 1.52. w-6 4.0 10 2048 1.15. l0-7 1.4 2.88 lo-s 1.6 5.75. 10-7 2.6 ITt 1000.0 Level .l a-1.0 Ci-0.5 a-0.0 llehllz ,, llehll2 lieh 112 0 2 4.98. 10 "2 2.77. 10 2 5.62. 10 _, 1 4 2.17. to-2 2.3 1.25. 10-2 2.2 4.12. w-2 1.4 2 8 6.40. 10-3 3.4 3.76. 10-3 3.3 1.66. 10-2 2.5 3 16 1.67. w-3 3.8 9.92. 10-4 3.8 5.36. w-3 3.1 4 32 4.27. w-4 3.9 2.52. 10-4 :l.9 1.52. 10-3 3.5 5 64 1.07. 10-4 4.0 6.34. w-s 4.0 4.09. to-4 3.7 6 128 2.69. 10-5 4.0 1.59. lo-s 4.0 1.06 10-4 3.8 7 256 6.75. 10-6 4.0 3.98. 10-6 4.0 2.10 1o-s 39 8 512 1.69. w-s 4.0 1.00 10-6 4.0 6.81. 10-6 4.0 9 1024 4.23. 10-7 4.0 12.52. 10-7 4.0 1.71. 10-6 4.0 I 10 2048 1.08. 10-7 4.0 7.42. w-s 3.4 4.27. 10-7 4.0 10 2048 7.28 lo-s 1.5 1 4.77. to-s 1.5 1.34. 10-7 3.2 Test problem: = q for (z, p) E [0, 1] x [-1, 1] } = 0 for JJ > 0 = 0 for I'< 0 where ITa=;:,, q := p1rcos(ou) +1Tasin(1rz), Exact Solution: lj;(z,p) = sin(1rz), Number of Moments: N = 4. 78

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T bl 4" M If "d a e ,.iJ: u 1gn convergence ac ors .or so vmg f t f I th t nations. e mornen eq (1,1)-V-cycle O"t 0! = 1.0 0! = 0.5 = 0.25 0! = 0.1 "'= 0.0 16 0.052 0.086 0.083 0.118 0.169 101 0.091 0.092 0.091 0.117 0.136 102 0.056 0.056 0.071 0.106 0.131 103 0.092 0.093 0.092 0.105 0.127 104 0.095 0 094 0.094 0.106 0.129 105 0.095 0.094 0.093 0.107 0.130 106 0.095 0.092 0.092 0.107 0.130 107 0.095 0.092 0.092 0.107 0.130 108 0.095 0.092 0.092 0.107 0.130 109 0.095 0.094 0.092 0.107 0.130 1010 0.095 0.094 0.092 0.106 0.130 (2,1)-V-cycle O"t ( = 1.0 ( = 0.5 ( = 0.25 ( = 0.1 ( = 0.0 10 0.074 0.051 0.054 0.105 0.155 101 0.055 0.055 0.055 0.082 0.104 102 0.02[1 0.025 0.039 0.077 0.097 103 0.023 0.026 0.042 0.076 0.096 104 0.023 0.023 0.042 0.076 0.096 105 0.023 0.023 0.042 0.076 0.096 106 0.023 0.023 0.042 0.076 0.095 107 0.023 0.023 0.042 0.076 0.095 108 0.023 0.023 0.042 0.076 0.095 109 0.023 0.023 0.042 0.076 0.095 1010 0.023 0.023 0.042 0.076 0.095 Test problem: = 0 for (z,l') E [0, 1] x [-1, 1] } = 0 for I'> 0 = 0 for I'< 0 where era= ..Q... o, Exact Solution: 1/J(z, !') = 0. Initial Iterate: randomly generated grid function. Mesh size: h = 1t8 Number of Moments: N = 8. 79

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Table 4.6: Full Multigrid (1,1)-V-Cycle convergence factors for solving the moment equa tions. IJt = 1.0 Level t = 1.0 "= 0.5 "= 0.0 llehll2 llehll2 lleh112 0 2 1.02 10 -l 2.45. 10 _, 2.84-10'1 1 4 3.71 10-2 2.7 7.66. 10-3 3.2 1.48-1 1.9 2 8 9.52. 10-3 3.9 2.22. Io-s 3.4 5.24. 10-2 2.8 3 16 2.38. 10-3 4.0 5.75. Io-4 3.8 1.55. 10-2 3.3 4 32 5.69 10-4 3.9 1.45 10-4 3.9 4.32. 10-3 2.0 5 64 1.49 10-4 4.0 3.62 10-5 4.0 1.15. 10-3 3.7 6 128 3.73. 10-5 3.9 9.07 Io-s 3.9 2.98. 10-4 3.8 7 256 9.33. 10-6 3.9 2.26 .. 10-6 4.0 7.62. 10-5 3.9 8 512 2.33. 10-6 4.0 5.66. 10-7 4.0 1.93. 10-5 :l.9 9 1024 5.83. 10-7 3.9 1.47. 10-7 4.0 4.86. 10-6 3.9 10 2048 1.45. 10-7 3.9 3.53. 10-8 4.0 1.22. 10-6 4.0 10 2048 1.00 10-7 1.4 2.68. 10-8 1.3 4.61. 10-7 2.6 IJt = 1000.0 Level -:r a= 1.0 "= 0.5 "-0.0 llehll2 llehll2 I, llehll2 0 2 4.31. 10-2 3.29. 10-3 4.74-10 2 1 4 1.83-2 2.3 4.09. 10-4 8.0 3.27. 10-2 1.4 2 8 4.93. 10-3 3.7 2.85. 1()-4 1.4 1.24. 10-2 2.6 3 16 1.23. 10-3 4.0 9.77. w-5 2.9 3.85. 10-3 3.2 4 32 3.09. 10-4 3.9 2.64. lo-s 3.7 1.07. 10-3 3.6 5 64 7.73. 10-5 3.9 6.78. 10-6 3.8 2.85 10-4 3.7 6 128 1.93 lo-s 4.0 1.71 10-6 3.9 7.36 10-5 3.8 7 256 4.83. 10-6 3.9 4.31. 10-7 3.9 1.87. 10-5 3.9 8 512 1.20. 10-6 4.0 1.08. Io-7 3.9 4.72-10-6 3.9 9 1024 3.02. 10-7 3.9 2.69. 10-8 4.0 1.18 lo-s 4.0 10 2048 7.56. 10-8 3.9 6.55. 10-9 4.1 2.97. 10-7 3.9 10 2048 4.48. 10-8 1.6 2.34. 10-9 2.8 9.32. 1o-s 3.1 Test problem: { [l't,+iJ,(l-P)+iJaP]if;(z,Jl) if;(O, I') if;(l, I') = q for (z,Jl) E [0, 1] X [-1, 1] } = 0 for I'> 0 = 0 for I'< 0 where I! a= 2.., q := vrrcos(7rZ) + o-asin(rrz), Exact Solution: ,P(z,Jl) = sin(1rz), Number of u, Moments: N = 4. 80

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CHAPTER 5 CONCLUSIONS 5.1 Summary of Results In this thesis, we have studied a systematic Least-Squares approach to the neutron transport equatio. The Least-Squares formulation converts the first-order transport problern into a self-adjoint variational form, which makes it accessible to the standard Finite-Element theory. Essential for this theory is the V -ellipticity and the continuity of the variational form, which leads directly to the existence and uniqueness of the analytic and discrete solutions and to bounds for the discretization error for a variety of different discrete space'S. Moreover, the variational formulation guides in a natural way the development of a multigrid solver for the resulting discrete problem. However) due to special properties of the transport equation, the Least-Squares approach is less straightforward than it first appears. In this thesis, we focused on neutron transport problems in diffusive regimes. In these regimes, the transport equation is singularly perturbed and its solution tends to a solution of a diffusion equation. Therefore, to guarantee an accurate discrete solution, a discretization for the transport operator is needed, that becomes a good approximation of the diffusion operator in diffusive regimes. Only a few conventional discretization schemes are known to have this property. By an asymptotic expansion, we show in Theorem 2.1 for slab geon1etry that a Least-Squares discretization with piecewise linear elements in space fails to be accurate in diffusive regimes. The choice of linear elements in space will for any right-hand side always result in a straight line connecting the prescribed values at the boundary for the principal part of the solution, which is independent of direction angle {t. Nurnerical tests confirm this behavior. On the other hand, we prove in Theorem 2.1 that, if piecewise polynomials of degree :2: 2 are used, then the principal part of the discrete Least-Squares solution becomes a Galer kin approximation to the correct diffusion equation in diffusive regimes. This means that the Least-Squares discretization will be accurate in this case. Numerical tests with piecewise quadratic elements again confirm this result. Because of Cea1s Lemma, the Least-Squares discretization can be viewed as the best approximation to the exact solution in the discrete space with respect to the Least-Squares nonn llfll, where f denotes the transport operator. In diffusive regimes, the different terms in the transport operator become totally unbalanced, which means that different parts of the solution are weighted much differently by the Least-Squares norm. With P denoting the L2-orthogonal projection onto the space of functions that are independent of direction angle, it is clear that the Least-Squares norm in diffusive regimes hardly measures the components P1p of the solution 1/;, although this is the main component for these regimes. The idea is therefore to scale the transport operator prior to the Least-Squares discretization, with the effect of changing the weighting in the Least-Squares norm_. Clearly, the scaling from the left by S = P + r(I-P) with r = 0(1/o-1 ) increases the weight for the important solution component P'lj;. Numerical tests show that a Least-Squares discretization of the scaled transport equation, even for piecewise linear elements in space, yields an accurate solution in diffusive regimes. Moreover 1 they show for piecewise quadratic elements that the

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scaling transformation dramatically increases accuracy. The major part of this thesis is devoted to proving that the Least-Squares dis cretization in combination with the scaling transformation S gives for a variety of simple Finite-Element spaces always accurate discrete solutions, even in diffusive regimes. As mentioned above, essential for bounding the error is the V -ellipticity and the continuity of the Least-Squares form with respect to some norm. [t is easy to show that the scaled Least Squares form cannot be bounded from below by a standard Sobolev norm. Therefore, the first obvious choice in the one-dimensional case is the norm (liP g, 112 + II II') 112. With respect to this norm, we prove V -ellipticity and continuity of the scaled Least-Squares bilinear form and derive error bounds for discrete spaces that use piecewise polynomials in space and piecewise polynomials or Legendre polynomials in angle as basis functions. However, since the V-ellipticity and the continuity constants for this norm depend on cr1 and era, these bounds blow up in diffusive regimes. To prove the V -ellipticity and continuity with constants independent of crt and cr a, we use a scaled norm. Based on the V -ellipticity and continuity with constants independent of CTt and cr a, we obtain discretization error bounds for the same discrete spaces mentioned above, with constants independent of crt and era. Thus, these bounds stay valid also in diffusive regimes. 'rhis result is generalized to three-dimensional x-y-z geometry for discrete space that use piecewise polynomials as basis functions in space and spherical harmonics as basis functions in angle. We conclude that the Least-Squares approach in combination with the scaling transformation represents a general framework for finding discretizations for the transport equa tion that are accurate in diffusive regimes. Further, it naturally guides naturally the development of an efficient rrmltigrid solver for the resulting discrete system. This is demonstrated in this thesis for slab geometry and piecewise linear elements. The developed multigrid solver for this discrete problem has convergence factors on the order of 0.1) so that one full multigrid cycle of this algorith1n computes a solution with an error on the order of the discretization error. 5.2 Recommendations for Future Work Our numerical results show that1 when simple discrete spaces in space are used1 refinement is needed in order to resolve boundary layers. 'I'herefore1 the aim for the future would be to combine the full multigrid solver with adaptive refinement. On the other hand, with the V -ellipticity and the continuity given, it seems fairly straightforward to establish error bounds for more complicated discrete spaces that can better resolve boundary layers1 including those of exponential or hierarchical type. Furthermore1 generalization of the scaling technique to anisotropic transport prob lems suggests itself. 82

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BIBLIOGRAPHY [1] R.A. ADAMS, Sobole1! Spaces, Academic Press, 1975. [2] R.E. ALCOUFFE, E.W. LARSEN, W.F. MILLER AND B.R. WIENKE, Computational bjjiciency of Numerical Methods for the Nfultigroup, Discre-te Ordinates Neutron Trans port Equations: The Slab Geomdry Case, Nuclear Science and Engineering 71, pp. 111-127, 1979. [3] G.B. ARFKEN, Mathematical Methods for Physicists, second edition, Academy Press, New York, 1971. [4] A. BARNETT, .J.E. MOREL AND D.R. HARRIS, A Multigrid Acceleration Method for the One-Dimensional SN .Equations with Anisotropic Scattering) Nuclear Science and Engineering 102, pp. 1-21, 1989. [5] J.H. BRAMBLE, Mttltigrid Methods, Pitman Research Notes in Mathematics Series 294, Longman Scientific and Technical, 1993. [6] W.L. BRIGGS, A Multigrid Tutorial, SIAM, Philadelphia, 1987. [7] C. BoRGERS, E.W. LARSEN AND M. L. ADAMS, The Asymptotic Diffusion Limit of a Linear Discontinuous Discretization of a Two-Dimensional Linear Transport Eq'uation, Journal of Computational Physics 98, pp. 285-300, 1992. [8] S.C. BRENNER, L.R. ScoTT, The Mathematical Theory of Pinite Element Methods, Texts in applied mathematics, Springer Verlag Inc., New York, 1994. [9] E. BRODA, Ludwig Boltzmann. JI!Ienseh. Physiker. Philosoph., Franz Deuticke Verlags gesellschaft m.b.H., Wien, 1986. [10] z. CAI, R. LAZAROV, T.A. MANTEUF'FF:L AND S.F. McCORMICK, Pirst-OrderSystem Least Squares for Partial Differential Equations: Part I, SIAM J. Numer. Anal., Vol. 31, 1994. [11] z. CAI, T.A. MANTEUFFEL AND S.F. McCoRMICK, Pirst-Order System Least Squares for Partial Differential Equations: Part II, submitted to SIAM J. Numer. Anal., March 1994. [12] Z. CAl, T.A. MANTEUFFEL AND S.F. McCoRMICK, Pirst-OrderSystem Least-Squares for the Stokes Equation, submitted to SIAM J. Numer. Anal., June 1994.

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[13] B.G. CARLSON AND K.D. LATHROP, Transport Theory-The Method of Discrete Ordirw-tes, in Cornputing Methods in Reactor Physics, (H. Greenspan, C.N. Kelber, and D. Okrent, eds.), Gordon and Breach, New York, p. 166, 1968. [14] K.M. CASE AND P.F. ZWEIFFEL, Linear Transport Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1967. [15] C. CERCIGNANI, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, Vol. 67, Springer-Verlag, New York, 1988. [16] P.G. C!ARLET AND J.L. LIONS, Handbook of Numerical Analysis, v. II, Finite Element Methods, Elsevier Science Publishers B. V. North-Holland, Amsterdam, 1991. [17] J.J. DUDERSTADT AND W.R MARTIN, Transport Theory, John Wiley & Sons, New York, 1978. [18] V. FABER AND 'r.A. MANTEUFFEL, Neutron Transport from the Viewpoint of Linear Algebra, Transport Theory, Invariant Imbedding and Integral Equations, (Nelson, Faber, Manteuffel, Seth, and White, eels.), Lecture Notes in Pure and Applied Mathematics, 115, pp. 37-61, Marcel-Decker, April 1989. [19] K.O. FRIEDRICHS, Asymptotic Phenomena in Mathematical Physics, Bull. Am. Math. Soc., 61, pp. 485-504, 1955. [20] G.H. GoLUB AND C.F. VAN LOAN, Matrix Computations, second edition, The Johns Hopkins University Press, Baltimore, 1989. [21] D. GOTTLIEB AND S.A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1977. [22] P. GR!SVARD, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Publishing Program, Boston, 1985. [23] G.J. HABETLER AND B.J. MATKOWSKY, Uniform Asymptotic Expansion in Transport Theory with Small Free Paths, and the Diffusion Approximation, Journal of Mathemat ical Physics 16, No. 4, pp. 846-854, Aprill975. [24] W. HACKBUSCH, Multi-Grid Methods and Applications, Springer, Berlin, 1985. [25] C. JOHNSON, Numerical Solution of Partial Differential Equations by the Fin-ite Element Method, Cambridge University Press, Cambridge, 1990. [26] S. KAPLAN AND J .A. DAVIS, Canonical and Involutory Transformations of the Varia tional Problems of Transport Theory, Nucl. Sci. Eng., 28, pp. 166-176, 1967. [27] J .R. LAMARSR, Introduction to Nuclear Reactor Theory, Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1965. 84

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[28] E.W. LARSEN, Diffusion Theory as an Asymptotic Limit of Transport Theory for Nearly Critical Systems with Small Mean Free Path, Annals of Nuclear Energy, Vol. 7, pp. 249255. [29] E. W. LARSEN, Diffusion-Synthetic Acceleration Method for Discrete Ordinates Prob lems, Transport Theory and Statistical Physics, 13, pp. 107-126, 1984. [30] E. W. LARSEN, The Asymptotic Diffusion Limit of Discretized Transport Problems, Nuclear Science and Engineering 112, pp. 336-346, 1992. [31] E. W. LARSEN AND J .B. KELLER, Asymptotic Solution of Neutron Transport Problems for Small Mean Free Paths, J. Math. Phys., Vol. 15, No.1, pp. 75-81, January 1974. [32] E.W. LARSEN, J.E. MoREL, AND W.F. MILLER, Asymptotic Solmions of Numerical Transport Problems in Optically Thick, Diffusive Re.qimes, J. Comp. Phys., 69, pp. 283-324, 1987. [33] E.W. LARSEN AND J.E. MoREL, Asymptotic Solutions of Numerical Transport Prob lems in Optically Thick Diffusive Regimes Il, J. Comp. Phys. 83, (1989), p. 212. [34] E. E. LEWIS AND W.F. MILLER, Computational Methods of Neutron Transport, John Wiley & Sons, New York, 1984. [35] T.A. MANTEUFFEL, unpublished personal notes on even-parity. [36] T.A. MANTEUFFEL, S.F. McCoRMICK, J.E. MoREL, S. OLIVEIRA AND G. YANG, A Fast Multigrid Solver for Isotropic Transport Problems, submitted to SIAM J. Sci. Comp., to appear. [37] T.A MANTEUFFEL, S.F. McCORMICK, J.E. MOREL, S. OLIVEIRA AND G. YANG, A parallel Version of a M-uUigrid Algorithm for Isotropic Transport Equations, submitted to SIAM J. Sci. and Stat. Comp. 15, No 2, pp. 474-493, March 1994. [38] T.A. MANTEUFFEL AND K.J. RESSEL, Multilevel Methods for Transport Equations in Diffusive Regimes, Proceedings of the Copper Mountain Conference on Multigrid Methods, April 5-9, 1993. [39] H. MARCENAU AND G.M. MuRPHY, The Mathematics of Physics and Chemistry, sec ond edition, D. Van Nostrand Company, Inc., Princeton, 1968. [40] W.R. MARTIN, The Application of the Finite Element Method to the Neutron Transport Equation, Ph.D. Thesis, Nuclear Engineering Department, The University of Michigan, Ann Arbor, Michigan, 1976. [41] S.F. McCORMICK, Mu/tigrid Methods, Frontiers in Applied Mathematics 3, SIAM, Philadelphia, 1987. 85

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[42] S.F. McCoRMICK, Multilevel Adaptive Methods for Partial Differential Equations, Frontiers in Applied Mathematics, SIAM, Philadelphia, 1989. [43] S.F. McCoRMICK, Multilevel Projection Methods for Partial Differential Equations, SIAM, Philadelphia, 1992. [44] J.E. MoREL AND T.A. MANTEUFFEL, An Angular Multigrid Acceleration Technique for the SN Equations with Highly Forward-Peaked Scattering, Nuclear Science and Engineering, 107, pp. 330-342, 1991. [45] J.T. ODEN AND G.F. CAREY, Finite Elements, Mathematical Aspects, Volume IV, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983. [46] ILK. OSBORN, S. YIP, The Foundations of Neutron Transport Theory, Gordon and Breach, Science Publishers, Inc., New York, 1966. [47] A.L PEHLIVANOV, G.F. CAREY AND R.D. LAZAROV, Least-Squares Mixed Finite Elements for Second-order Elliptic Problems, SIAM J. Nurner. Anal., Vol. 31, No.5, pp 1368-1377, October 1994. [48] G.C. PoMRANING, Diffusive Limits for Linear Transport Equations, Nuclear Science and Engineering 112, pp. 239-255, 1992. [49] J. STOER AND R. BULIRSCH, Introduction to Numerical Analysis, second edition, Texts in applied mathematics1 Springer Verlag1 New York1 1993. [50] S. UKAI, Solution of Multi-dimensional Neutron Transport Equations by Finite Element Methods, Journal of Nuclear Science and Technology, 9(6), pp. 366-373, 1972. [51] G.M.WING, An Introdaction to Transport Theory, John Wiley and Sons, Inc., New York, 1962. 86

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APPENDIX A FLUX STENCIL In this section, we derive the stencil for the Least-Squares discretization of the SN fiux equations (4.16) for piecewise linear elements. VVe assume that the slab is partitioned into ZJ = z0 < z1 < < Zm = Zr and denote by hk := Zk -Zk-1 the cell width of cell k. We are then looking for a discrete solution in the form m N m
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,, Zi+l J (niL<, IL'l,,;,j) JRN dz +j ( niL,P, !Lry1 +1 ) dz -,1 ,J JRN Zi-1 ,, (*1) (*2) (A.3) Zi+l J ( Oq !Lry ) dz -S -r,t,) JRN +[ (nq ,IL'L 1 ) dz -s ,z+ ,J JRN Zi-1 ,, (*B) (*4) In the following, we consider the terms (*1) to (*4) separately. To (*1): Applying the substitution z = zZi-11 we have !,, I nJL,;;, ILr1 .. ) dz = jh' I niL{/;, ILr1 . ) dz \ -r,z,J JRN \ --'4,) JRN z,_l 0 where h = hi and z !lr,j hf..j, with 11 := 1/:k_1 and 1, := <,. Then it follows that fL1J.,,j [1w T + T(J -1w T)] Mr; + [m,(l1w T) + O"alw TJ Zfj !L< [1w T + T(J-1w T)j M (', -<1 ) + [m,(llw T) + o-,lw TJ (<1(hz) + J:,z) 88

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so that h J (mL;p, JL,7 .) dz -r,J fffN 0 + ( [r
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get the following contribution from (*1): h; [ 2 2 n ( 2 2 2 ) T] } +6 71: O't,iH + CJa,i 7i crt,i ww '!fi-l h; [ 2 2 ( 2 2 2 ) Tl} +TiCTti0+ (Jai-TiCTti WW 1/J .. 3 ) -t To (*2): Applying the substitution z = z -Zi, we have h;+l J ( 11ILio_, lL']_1J JRN dz, 0 where now ( h := h.;+!) and h-z '!L,j -h-f.j Therefore, ILry, ,J [1w T + r(Ilw T)] lvifj + [nr,(I -1w T) +
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so that h J I rw>0 ,ILr;, ) dz \ ,J JRN 0 + ( [ m,(I lw T) + oalw TJ 1/;_1 [ro-r(I lw T) + oa lw TJ 'Cj) JRN + ( [rcr,(Ilw T) + D"alW TJ 1/;_,., [m,(J-lw T) + D"alW TJ fj) JRN = --,1 (o [MlwT +r2M(I -lwT)] M (p_, -p_,) \0 (tTaMlw T + r2o-,M(Jlw T)j (p_, + 1/;_,) fj) JRN Consider all possible j and recall that h = hi+! and that (a-a, tTt, r) are the corresponding values in cell i + 1, denoted by ( oa i+l, a-, i+l, Ti+!), and that 1p1 = 1/J ., 1j; = 1j; We get , --t -'-T -t+l the following contribution from (*2): 91

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To (*3): We have q (z) = SNq(z) = lw T q(z) + r(IlwT)q(z), --where Therefore, Jz; / rlq .. ) dz \ fftN Zi-1 ,, = j (mw T 'l_, [ll;;_T + r(Ilw T)] MjrJ;,,)n 1 N dz z, + j (mw T 'l_, [nr,(Jlw T) + o-alw TJ f.j'lr,i) JRN dz X; + j (nr(I -lw T)'l_, [lw T + r(Ilw Tl] dz Xi-1. '' + j (rlr(I -lw T)'l_, [w,(I -lw T) + o-alw TJ f.jr),,i)JRN dz. 92

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Considering all possible j and recalling that r, crt, era are the corresponding values in cell i, we get with z; j 'J/}c,idz '= Zi-1 ( J:.'_, q(z, l'lh,;(z)dz ) !,','_, q(z, the following contribution from (*3): z; [(1r;') + rlOM] J 'l_'l;,idz Zi-1 + [rlo-t,iO + (o-a,irlo-t,i) ww T] j 'l_''lc,idz. To (*4): Similarly it follows that z,+l J (Oq ILry, '+l ) dz ,J JRN Z,+l j (mwTg_,[lwT +r(I-lwT)]Mf;'l;,i+l)11wdz Zt+l+ J (mw T '1_, [ro-,(J-Jw T) + O"alW Tl fj'!l,i+l)JRN dz z,+l + j (Or (Ilw T)g_, [lw T + r(Ilw T)] Mr;'l:,i+l) JRN dz z,+l + j (Or (Ilw T)g_, [ ro-,(Ilw T) + o-a lw TJ r;r/l,i+l) JRN dz. Considering all possible j and recalling that r, crt, era are the corresponding values in cell 93

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i + 1, we get with Zt+l j lj_'ll,i+l dz ,, the following contribution from (*4): Zi+l j lj_'l;,i+l dz ,, Zi+l + + (aa,i+l-rl+10"t,i+1) ww TJ j 9_1JI,i+ldz. ', 94

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Equations for an interior node Putting everything together, we get the following equations for an interior node i: + { [(r;2+1 -1) Mww T Mr;'+10M2 ) + [(r;2 + 1
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Equations for the left boundary For the left boundary ( i = 0 ), we have only contributions from part (*2) and part (*4) and, in addition, j runs only from + 1, ... N, since the values for the positive angles are given by the boundary conditions. With MN/2 := diag(p,, ... !'N/2); nN/2 := diag(w,, ... ,WNj2); iNj2 := (w,, .. 'WNj2) T and with for the left boundary the following equations we get { [(1r1 2 ) Jvrww T M + rfnM2 ] h1 [ 2 2 ,1 ( 2 2 2 ) Tl} ,1, +3 + O'"a,l-710"t,l La { 1 [( 2 ) T 2n-M'] + -r1 -1 M ww M r1 h, h1 [ 2 2 ,1 ( 2 2 2 ) Tl} .1. +6 + O'a,l-710''1,1 1'..1 ,, = [(1r?) M-ww T + r{W M] j g_ry;,1dz '" + [rf,.,,,n+ (,.a,l-r{,.,,,) iN;ziT] j 'l_ry1,1dz. 96

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Equations for the right boundary For the right boundary ( i-m) we have only contributions from part (kl) and part (*3) and, in addition, j runs only from 1, ... If, since the values for the negative angles are given by the boundary conditions. With n+ := ( nN/2 0 ) we get the following equations for the right boundary: hm [ 2 2 n+ ( 2 2 2 ) T l } /, +3 rmcrt,m + 17a,m-7ml7t,m :t.m ''" [(1r,;,) M+ww T + r,;,n+ M] j 'I'I;,mdz Zrn-1 + [r,;,,,-,,mn+ + (o-,,m-r,;,o-,,m)\!I.N;2"'T] l f'lc,mdz. Zm-1 97

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APPENDIX B MOMENT STENCIL In this section, we list the stencil for the Least-Squares discretization of the moment equations (4.18) for piecewise linear basis functions. Assuming that each moment tPz(z) is of the form and defining m
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Stencil for Stencil for 1'k,l with l > 1: [ 2 -]] h-2h-1 r, h;:
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we can write all the above equations as a system of the form Ao,o Ao,l ( Al,O A1,1 A1,2 'lo ) A =: q. (B.l) Am-l,m-2 Am-l,m-1 Am-1,m !]_rn A.m,m-1 Am,m J So far we have not taken into consideration the boundary conditions (4.14), which can be written as J=(t)=g_, (B.2) with [ [h, 0]1'T 0 0 ] ( gl(/h) ) ( g,(I'J;j-H) ) J := 0 0 [0 IN ]TT ; ':l, := : ; 'l, := : 0 Nx(N(m+l)) m(I'J;j-) g,(!'N) We have therefore to minimize the Least-Squares functional F( ) subject to the constraint (B.2). This can be done by making the total derivative ofF zero, which we write as [iJ] iJ \7 <;F( ) = 0, (B.3) where denotes the m.atrix '] .. Because of the constraint (B.2), the F( ) = 0 results in the discrete system (B. 1 ). Forming derivatives /! of the constraint o/k,! (B.2) gives [iJ] T J iJ = 0; therefore, all rows of are in the nullspace N(J), which has dimension Nm. Suppose we find an ( N m) x N matrix J J. of rank N m such that J ( J J.) T = 0. 'I' his means then that the rows of J J.span N( J). Therefore, there exists a matrix D with D [ = J J.. Multiplying ( 8.3) by D results in J J. \7 .,F( ) = 0, which together with the constraint becomes the closed system { ;:cv,F(). = o } J = H_ Since TT TO = IN, then we can choose J J.. in the following way: with 0 IN(m-1) 0 ] G, [ J N l c, := nl (J j 100 (B.4)

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Taking into account that 'Vi!>F(A= 'I' we see that (B.4) can be written as Co,o Co,l qo Al,O A1,1 A,,, <1>= (B.5) Arn-l,m-2 Am-l,m-1 A.m-l,m l[m-1 Cm,m-1 Cm,m ijm with [ [I1,0]TT] I 0 ] [ !!., l Co,o Co,t l G"[ Ao,J ij_o J G[ Ao,o GJ:l [ GJAm,m ] [ cr A:,m-1 ] [ Cm,rn .Cm,m-1 -[O,h]TT 101