The Three-field Formulation for Elliptic Equations:
Stabilization and Decoupling Strategies
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Dissertation
zur Erlangung des Doktorgrades
der Mathematisch-Naturwissenschaftlichen Fakultäten
der Georg–August–Universität zu Göttingen
vorgelegt von
Gerd Rapin
aus Haselünne
Göttingen 2003
D7
Referent:
Prof. Dr. G. Lube
Korreferent:
Prof. Dr. R. Schaback
Tag der mündlichen Prüfung:
16.7.2003
Contents
Introduction
1
I The single domain problem
5
1
The advection-diffusion-reaction problem
7
2
Imposing Dirichlet conditions in a weak sense
2.1 The weak formulation . . . . . . . . . . .
2.2 Discretization and stabilization . . . . . .
2.3 Stability . . . . . . . . . . . . . . . . . .
2.4 A priori analysis . . . . . . . . . . . . . .
2.5 A posteriori analysis . . . . . . . . . . .
2.6 Numerical results . . . . . . . . . . . . .
2.7 Conclusions . . . . . . . . . . . . . . . .
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11
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II The three-field formulation
39
3
The three-field formulation
3.1 The continuous three-field formulation . . . . . . . . . . . . . . . . . . . . . . .
3.2 The discrete version of the three-field formulation . . . . . . . . . . . . . . . . .
3.3 Connection to mortar elements . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
42
51
54
4
A stabilized three-field formulation
4.1 A discrete stabilized scheme . .
4.2 Analysis of the stabilized scheme
4.3 Numerical results . . . . . . . .
4.4 Conclusions . . . . . . . . . . .
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57
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68
76
The three-field formulation for the Oseen Equations
5.1 The Oseen equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The three-field formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
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ii
CONTENTS
III Nonoverlapping domain decomposition methods
6
7
85
A preconditioned Schur complement method
6.1 The continuous case . . . . . . . . . . . .
6.2 The discrete case . . . . . . . . . . . . .
6.3 Numerical results . . . . . . . . . . . . .
6.4 Conclusions . . . . . . . . . . . . . . . .
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87
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An alternating Schwarz algorithm
7.1 The continuous formulation . .
7.2 Discretization . . . . . . . . .
7.3 Numerical results . . . . . . .
7.4 Conclusions . . . . . . . . . .
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111
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8
Comparison of some nonoverlapping domain decomposition methods
123
8.1 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9
Summary and Outlook
141
IV Appendix
143
A Functional Analysis
145
A.1 Some basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.2 Closed Range Theorem and applications . . . . . . . . . . . . . . . . . . . . . . 147
B Function spaces
B.1 Smooth functions . . . . . . . . . . . . . . . . . . . .
B.2 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . .
B.3 Distributions, weak derivatives and Sobolev spaces . .
B.4 Trace theorems and Sobolev spaces of fractional order
B.5 Some fundamental equalities and inequalities . . . . .
B.6 Finite Element spaces . . . . . . . . . . . . . . . . . .
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153
153
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157
159
160
Bibliography
165
Notation
173
Index
179
Introduction
This thesis deals with domain decomposition methods for advection-diffusion equations. In the
last years domain decomposition methods have become a very active research area in the field
of the numerical approximation of partial differential equations. The key idea of domain decomposition methods is simple to explain: The global boundary value problem, given in a domain
, is divided into local boundary value problems in subdomains , the union of which gives
(cf. Figure 1). The local problems are linked together by suitable coupling terms or transmission
conditions. This leads to discrete schemes like the mortar method (cf. [BMP94, Bel99, Woh99])
or the three-field formulation (cf. [BM94, BM92]). The more general latter approach is presented
here.
Domain decomposition methods allow to couple different models, i.e. different partial differential
equations or different discretization methods on local subdomains. In this work we concentrate on
finite element discretizations given in local subdomains. Each discretization can be independent
of the remaining ones (cf. Figure 1 (b)). Therefore we are interested in the case of nonmatching
grids, which causes nonmatching ansatz functions on the boundaries of the subdomains. By virtue
of our approach it is possible to apply different software tools for specific geometries on complex
domains by dividing the domain into subdomains with these specific geometries.
Having such a multi-domain formulation there are several strategies to split the global problem into
a sequence of local problems by iterative decoupling. Assigning the local problems to different
processors we get a very intrinsic way to solve our numerical problems in parallel. In complex
three-dimensional domains the use of parallel methods is mandatory.
The resulting methods can be classified into several groups. First it can be differentiated between
nonoverlapping and overlapping methods. In the overlapping case the domain is divided into
overlapping subdomains . The alternating Schwarz method, introduced by H.A. S CHWARZ
in 18691 , was probably the first example of a domain decomposition method. Starting with a
and (cf. Figure 1 (a)) the equations are
decomposition into two overlapping subdomains
solved iteratively on the subdomains using Dirichlet values of the neighbor domains computed
in the previous step. In this way H.A. S CHWARZ could show the existence of a solution of the
Poisson problem for a domain with nonsmooth boundary.
Moreover one can distinguish between additive and multiplicative
Schwarz methods. Denoting
by for the two-domain
the solution of iteration step in subdomain
case the multiplicative
is
variant can be described as follows: Starting with an initial guess, first a new solution in
computed. Then, already using this solution, the solution in
is solved, and so on. In contrast
the additive algorithm uses the solution of the previous step instead of the current solution (cf.
Figure 2). The second method has got the advantage that the solution of all subdomain problems
1
cf. O.B. W IDLUND [Wid90] for a short history of domain decomposition methods
2
Introduction
Ω
Ω1
Ω2
Ω2
Ω1
Ω
(a) The original example of H.A. S CHWARZ
(b) Decomposition into simple domains
Figure 1: The figure shows two simple decompositions. (a) is an overlapping decomposition. In (b) the
meshes of and are nonmatching at the interface.
can be completely done in parallel. In the multi-domain case the multiplicative variant requires a
coloring of the subdomains.
In this thesis we focus on nonoverlapping methods. Overlapping methods have the drawback of
some overlap of data and very often the partitions are much harder to generate. Moreover, different
models in different subdomains require the nonoverlapping approach.
A direct analogue of the Schwarz algorithm to the nonoverlapping case is not possible, because
in general the iterative scheme does not converge, if Dirichlet data of the subdomain boundaries
is interchanged. But if we replace the Dirichlet-condition by other transmission conditions like
Neumann- or Robin-conditions, we get further classes of methods, sometimes called iterationby-subdomain methods. This leads to schemes like the Robin-Robin (cf. [LMO00, NR95]), the
Dirichlet-Neumann (cf. [GGQ96]) or the Robin-Dirichlet (cf. [ATV98]) method. To demonstrate
these methods the interchanging of Robin conditions across the interface is discussed in this work.
A second well established class of methods, called iterative substructuring methods, is given by
a linear system for the interface degrees of freedom, which is constructed by eliminating the unknowns inside the subdomains. On the discrete level the resulting equation is called the Schur
complement equation; on the continuous level the equation depends on the Steklov–Poincar é operator. Applying the Steklov–Poincaré operator resp. the Schur complement matrix corresponds
to the solution of local problems with Dirichlet conditions on the interface. Normally the discrete equation is solved by an iterative algorithm. Especially Krylov methods like CG or GMRES
methods are used, where each step requires the solution of local boundary value problems. Since
the interface equation is poorly conditioned, preconditioning is essential for an efficient implementation. The construction of good preconditioners for the Schur complement equation is a very
active research area. In order to be able to parallelize the solution procedure, the preconditioners
are built by local problems. So we get for example the BPS-preconditioner (cf. [BPS86]), the
Neumann-Neumann preconditioner (cf. [DW95, DSW94]) or the Robin-Robin preconditioner (cf.
[AJT 99, ATNV00]). A variant of the latter preconditioner is presented in chapter 6.
In this work we try to give a unified presentation of some nonoverlapping domain decomposition
methods for the stationary advection–diffusion–reaction equation
3
additive Schwarz algorithm
1. initial guess , 2. 3. until convergence
4.
, 5.
Compute using 6. end
multiplicative Schwarz algorithm
1. initial guess 2. 3. until convergence
4.
5.
Compute using 6.
Compute using 7. end
Figure 2: Additive and multiplicative Schwarz algorithm for two subdomains.
in a bounded domain . The starting point of the analysis is a variant of the three-field formulation
of F. B REZZI and D. M ARINI (cf. [BM94, BM92]). Given a partition of into subdomains
three different classes of function spaces are defined. The first one lives on the local subdomains,
the second one is a space of Lagrange multipliers defined on the local boundaries of the subdomains and the third one is given on the union of the local subdomains, called (global) interface.
If these spaces are coupled by specific terms, we get an alternative, well posed, hybrid problem.
This formulation is treated in chapter 3.
A direct discretization of this scheme requires, that two conditions, called Babuška-Brezzi conditions, are satisfied. The mathematical treatment of the arising saddle point problems is briefly
discussed in the appendix. The first one demands that the function space of the local functions is
sufficiently ’rich’ compared to the space of Lagrange multipliers. In contrast the second inf-sup
condition requires the same relation between the space of Lagrange multipliers and the third class
of functions. But because the discrete ansatz spaces should be chosen completely independent, in
this work the Babuška-Brezzi conditions are circumvented by adding some stabilization terms.
Further difficulties arise in the singularly perturbed case, the case of . Therefore we introduce the SUPG-method in the local subdomains in order to suspend oscillations in streamwise
direction. Together with the above stabilization terms we get a new stabilized three-field formulation. Its analysis is discussed in chapter 4. An a-priori result is derived in special consideration of
the singularly perturbed case and is used to determine certain stabilization parameters. Our results
are optimal compared to the standard SUPG-method.
When using this approach on the local subdomains local Dirichlet problems arise in an intrinsic
way. The boundary conditions are worked in with the help of Lagrange multipliers. Since the arising local systems are interesting by themselves (fictious domain approaches, wavelet discretizations), we investigate them in detail in chapter 2 and derive a priori and a posteriori estimates. So
far in the literature these schemes have not been extended to the nonsymmetric case nor extensive
numerical studies have been carried out. Here, we will close this gap.
In a next step it is shown, that the stabilized three-field formulation is a proper basis for a unified
presentation of nonoverlapping methods. This is demonstrated on the continuous and the discrete
level for two typical algorithms in part III of the thesis.
In chapter 6 the Schur complement equation is derived from the three-field formulation. As a
preconditioner we use a proposal of Y. ACHDOU ET AL . (cf. [AJT 99, ATNV00]). The preconditioner is built up by solving local boundary value problems with Robin conditions on the interface.
Unfortunately the analysis of this method is not complete. Because of the nonsymmetric structure
4
Introduction
of the problem the standard techniques for symmetric problems cannot be applied.
In chapter 7 an iteration-by-subdomain algorithm is derived following a technique of
R. G LOWINSKI and P. L E TALLEC (cf. [GT89, GT90]). So we get an algorithm, where in
each iteration step Robin conditions at the local interfaces are interchanged. Finally, both methods
will be compared by some numerical experiments and by a Fourier analysis for the case of two
subdomains and constant coefficients (cf. G. R APIN, G. L UBE [RL01]).
Moreover in chapter 5 it is explained, how the three-field formulation can be extended to the Oseen equations. The presence of the pressure and the divergence-free constraint cause additional
difficulties. This is the first attempt of such an extension. The Oseen equations arise in many
linearization strategies of the Navier–Stokes equations. Therefore, normally a huge amount of
degrees of freedom is used in order to resolve the finer scales of the solution. Hence, parallel
methods for the Oseen equations are very important.
The thesis is split into four parts. In the first part we introduce the advection-diffusion-reaction
equation and discuss weakly enforced Dirichlet conditions for a single domain. The second part
is dedicated to the three-field formulation and includes the chapters about the stabilization and
the extension to the Oseen equations. In part III we show, how the three-field formulation can be
solved efficiently by iterative decoupling. We present two different algorithms and compare their
performance.
We complete this work by an appendix, where the functional setting and some auxiliary results
are presented: In appendix A some basic results of functional analysis are cited and the theory
of saddle point problems is developed. Then the definitions and properties of different function
spaces, which are used, are shortly reported in appendix B. Finally, we give a brief introduction to
the theory of finite element methods.
Acknowledgments
First, I would like to thank my adviser Prof. Dr. G. Lube for his kind assistance and support in
writing this thesis. His revisions and advices were always a great help for me.
Moreover I am very grateful to Prof. Dr. C. Canuto for his valuable suggestions, many discussions and his remarkable hospitality. During my stay at the Politecnico Torino in spring 2002 I
gained a deep insight into the Italian way of life. For the financial support, I wish to thank the
’Graduiertenkolleg für Strömungsinstabilitäten und Turbulenz’.
It is a great pleasure to express my gratitude to Prof. Dr. P. Hähner. The thesis benefits strongly
from his comments and advices. I am very grateful to T. Knopp for reading over parts of the thesis
for correct use of the English language. Furthermore, I am extremly grateful to Uta Engels for
her patience and inspiration. Sometimes she had a rather difficult time with me, when I tried to
concentrate on my work.
I would also like to thank the other members of the Institute of Numerical and Applied Mathematics for many fruitful discussions. In particular, I profited by the excellent support of the system
administrators Dr. G. Siebrasse, R. Wassmann and J. Perske.
Furthermore, I thank my parents and all my friends, who always remind me of the existence of a
life beyond the mathematics.
Part I
The single domain problem
Chapter 1
The advection-diffusion-reaction
problem
In this chapter we introduce the advection-diffusion-reaction problem. Our interest in this problem is particularly motivated by applications in science and economics. For example the equation
appears in computional fluid dynamics (CFD), chemistry, or in financial modeling (Black-Scholes
model). A nice overview about some important applications is given in the book of K.W. M OR TON (cf. [Mor96], ch. 1.1). The problem can also be considered as a simplified model of the
linearized Navier-Stokes equations. The extension of the presented numerical schemes to these
equations is briefly discussed in chapter 5.
In the following chapters some nonstandard numerical schemes for this equation are considered.
As already mentioned in the introduction we will mainly focus on a modified, stabilized three-field
formulation and different domain decomposition methods. But we also derive a new scheme for
the single domain where inhomogeneous Dirichlet conditions are enforced weakly.
The treatment of this problem has to be done properly because the solutions can possess sharp
layers. Therefore the extension of existing methods for symmetric problems to the advectiondiffusion problem is often not straightforward.
, , be a bounded, polyhedral domain with Lipschitz boundary
. For simplicLet
ity we impose homogenous Dirichlet conditions on the boundary
for a moment. Afterwards
we will also allow inhomogeneous boundary conditions. But first let us discuss the following
boundary value problem:
in
on
(1.1)
with diffusion coefficient , a given flow , source term , and reaction coefficient . The
following regularity of the data is required:
Additionally we assume
with
! " $#
a. e. in
%#
(Ass. 1)
(Ass. 2)
8
The advection-diffusion-reaction problem
Sometimes the stronger condition
a. e. in
(Ass. 2a)
is imposed. Then the variational formulation of (1.1) is given by
(1.2)
. As usual the weak formulation is obtained by integration
Find for a domain and #
by parts of ! The bilinear form and the linear form of the weak formulation (1.2) have the following properties, where we use
the
usual notation for the different Sobolev norms (cf. appendix, chapter B).
Especially the norm is denoted by "#" .
Lemma 1.1 Let be a domain and (Ass. 1), (Ass. 2) resp. (Ass.
2a) be valid. Then the
bilinear form # is continuous, i.e. there exists a constant $ such that
&% $ " '" ( ")*" ( $#
(1.3)
Further there exists a constant depending
on such that
" '" ( $#
,+
Additionally belongs to .
In order to simplify notation we will use the notation -/. , when there exists a constant 0 ,
% 0 . . For example
independent of and other important quantities like the mesh size, such that the inequality (1.3) can be written as
- " 1" ( ")2" ( #
Analogously 435. is defined. If 6-5. and 635. hold, we write 675. .
Proof: Let be . The continuity is obtained by virtue of the generalized Hölder and
the Cauchy-Schwarz inequality
8% % :9 ;=< " " " " ?> " @"BACED GF " 1" ")*" &
; (
H< " . " A IJD GF ")*" A IKD GF - L M" "BACED GF " N" DPORQ)D GFSF=TVU " '" ( ")*" ( with
9
where we have used Theorem B.4 in the last step. Furthermore, using integration by parts and the
inequality of Poincaré (Theorem B.12) we get
(
3 " 1" ( #
Finally, the assertion
+
(1.4)
follows similarly.
Remark 1.1 Note,
that Lemma 1.1 is also valid for weaker assumptions for . It is sufficient, to
impose .
Then with the help of the Lax-Milgram Lemma (Theorem A.2) we can prove, that the boundary
value problem (1.2) is well posed:
Lemma 1.2 There exists a unique solution of (1.2).
In the context of domain decomposition methods the following extension operator is very important. Sometimes this extension is called elliptic extension.
Q Corollary 1.1 Suppose a Lipschitz domain
with a piecewise smooth
manifold
is given. Then, for an arbitrary
there exists a unique
and
#
& satisfying
Furthermore, the a priori estimate
( %
0
" " 0 " " O Q D F
holds true.
This extension is often used in this work and will be denoted by
#
Proof: The proof of existence and uniqueness follows from the Lax-Milgram Lemma. The a
priori estimate can be found in P. G RISVARD [Gri85].
0
on causes severe problems. It reflects the dependence of the solution
The dependence of
of (1.2) on the diffusion coefficient . If is small compared to the advection and the reaction
coefficient, called the singularly perturbed case, the solution behaves almost like the solution of
the reduced problem, which is given by
with function space
Find ! #
" M on
(1.5)
10
The advection-diffusion-reaction problem
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
−0.2
1
0.6
0.8
0.6
0.8
0.4
0.2
0
1
Figure 1.1: The solution of (1.6) for and the solution of the limit problem are printed in one plot.
The upper solution without boundary layer is the solution of the limit problem. One can observe that the
boundary layer is very narrow.
where
is the inflow of the boundary. Therefore sharp layers
may occur. Layers are very narrow regions, where the solution or its derivatives change abruptly.
The correct numerical treatment of these layers in the interior has been widely analyzed in the
last twenty years, cf. [RST96]. Analyzing the layers near the boundary is the subject of the next
chapter.
Finally, in order to illustrate the dependency between the elliptic solution and the solution of the
hyperbolic limit, the following example of G. M ATTHIES , L. T OBISKA [MT01] on the unit square
will be briefly discussed:
The right hand side
in
on
"#
(1.6)
and the Dirichlet boundary condition are chosen such that
#
and the solution of the limit probbecomes the exact solution. In Figure 1.1 the solution for lem are plotted. We observe a different behavior of the solutions close to the outflow boundary.
There the elliptic solution possesses a strong boundary layer caused by the boundary conditions.
In contrast the hyperbolic solution is smooth close to the outflow boundary.
Chapter 2
Imposing Dirichlet conditions in a weak
sense
In this chapter boundary value problems with inhomogeneous boundary conditions are considered. The boundary conditions are not treated in the usual way; instead they are worked in with
the help of Lagrange multipliers. It will be figured out, that the Lagrange multipliers are given
by the approximation of the normal derivative of the searched solution. In many applications the
normal derivative is of great interest.
Imposing essential boundary conditions by Lagrange multipliers was first introduced and analyzed
by I. Babuška [Bab73]. He derived optimal convergence results under the restriction that the finite
element spaces satisfy a certain inf-sup condition, sometimes called Babuška-Brezzi condition.
J. P ITK ÄRANTA [Pit80, Pit79] showed that therefore the finite element spaces have to be chosen very carefully. By adding some stabilization terms H.J.C. BARBOSA and T.J.R. H UGHES
[BH92] could derive a stable scheme, where the different finite element functions could be chosen
completely arbitrary. Later this scheme was simplified by R. S TENBERG [Ste95]. R. S TENBERG
has also shown the close connection of this method to Nitsche’s method for solving Dirichlet problems (cf. [Nit71]).
Lately, treating boundary conditions in such a manner has become of great interest. For example
it is used in the context of wavelet discretizations (cf. [DK01], [Ber00b]) or in nonconforming
domain decomposition methods (cf. [BBM92, BM01], [TS95], [BK00]), where we typically work
with different nonmatching grids on the skeleton.
But up to now nearly all approaches have been restricted to the symmetric case. Here the scheme
, ,
is extended to the advection-diffusion problem in bounded Lipschitz domains
i.e.
! %#
on in
(2.1)
Q (2.2)
For the data we require that (Ass. 1) is satisfied and for the boundary data is assumed.
In this work we are particularly interested in the singularly perturbed case. Therefore, using ideas
of discontinuous Galerkin methods (cf. [Fre01, BO99, HSS02, SSH00]), we propose a new stabilized scheme and derive an optimal a priori estimate and an a posteriori error estimation for it. Our
method has been inspired by the work of C. S CHWAB, E. S ÜLI and P. H OUSTON. But in contrast
to their work we have to take care of the Lagrange multipliers.
12
Imposing Dirichlet conditions in a weak sense
We consider two variants for the design of stabilization parameters in the singularly perturbed
case: The first variant gives control of the Lagrange multiplier for the perturbed problem with
whereas the second variant approximates the corresponding term for the limit problem with
.
The latter case is of interest when the method is applied to nonoverlapping domain decomposition
methods. For example in Schur complement methods local problems with Dirichlet conditions on
the interface have to be solved iteratively. Typically the Dirichlet conditions on the interface are
given by previous iteration steps and therefore on the outflow part of the interface sharp boundary
layers may occur. Converging to the correct solution these layers usually become smaller. Avoiding these boundary layers by taking the latter choice of the stabilization parameters will give us
fast convergence, because for small the searched solution can be approximated quite well by the
solution of the reduced problem. Therefore in the context of domain decomposition methods it is
often a better strategy to approximate the reduced problem on the outflow than the elliptic solution
for . In chapter 3 this scheme will be applied to the three-field method for advection-diffusion
equations (cf. [RL03]).
!
2.1 The weak formulation
+
Q
L
U
To this end we define the space of the Lagrange multipliers with norm " " .
Q
( of . ForGF a domain the Sobolev
The space is given as the( dual of the trace space (
are denoted by " " /" " D . In( the case we
norms of the spaces (
simply write " &"
and the corresponding scalar product is given by . The norm
Q
is also denoted by " G" . The
dual
product
of
is
written
by
. Sometimes the
same notation is used for
is used. scalar products. For functions , the notation
Then the weak formulation of (2.1), (2.2) is given by
Find # (2.3)
Let us start with a slightly modified version of the variational formulation of I. BABU ŠKA [Bab73].
!
"
#
For simplicity we assume the condition (Ass. 2a) to ensure the well-posedness of (2.3). With
small modifications it would be also possible to prove the main result under the weaker condition
(Ass. 2).
We observe, that the problem (2.3) can be interpreted as a linear mixed problem or saddle point
problem. Therefore the theory of mixed problems can be applied (cf. Theorem A.4). A detailed
description of the theory can be found in the appendix.
% + % + and of , i.e.
")*" $#
Theorem 2.1 Let
and $ be real Hilbert spaces. Furthermore, let %
,+
+
&
$
be linear, bounded
operators where % is elliptic
on the kernel )
*%
.-0/21-
Then, if the Babuška-Brezzi
43 condition
6 5
687:9<;=
-
B 7 C
; =
>@?- A >
*(
)
")*" " " DE 9 / 1 9
- 9
'&
*(
(
(
*(
3
(2.4)
2.1 The weak formulation
13
is satisfied, the linear mixed problem
%
+ , +
+ *
Find
(
(
$
in
in
++
$
(2.5)
$
with
" 1" " 1" % 0 " " " " "2" #
(2.6)
+
In our situation the operator % is defined by
+ % by the bilinear
and the operator
form . Then, assuming the
boundary is Lipschitz, the kernel is given by (cf. Theorem B.7).
with $
+ 3
possesses a unique solution
-
(
*(
%
9
*
%
#&
9 /
-E/
DE / 1 #
&
)
*(
To apply Theorem 2.1, among other things, we have to show that the inf-sup condition is fulfilled.
We include the short proof, because we will use this result later for the a posteriori estimation.
0 not depending on , such that
#
(
0
F
O Q D ")*" " "
Lemma 2.1 There exists a constant
5
6 7 ; =
BC7
>
;=
?A
>
Q .
0 " " ( % 0" " O Q D F Q (cf. Theorem B.9). Then for we obtain the assertion by computing
%
" ?" O Q D F "1" O Q D F 0 O Q D F " 1" ( #
% 0 F
ORQ)D ")2" ( Proof: For the proof we choose a linear, bounded extension operator
Hence, there is a constant
such that
7
;=
?A
!
7
>
B27
;=
?A
Next the ellipticity of % on ) *(
is the energy norm for
and
by
mains
" 2" ?A
;=
!
>
>
is a consequence of Lemma 1.1. " "J . The natural extension of this definition to do(
" K 2" will be used further on. Then, taking into account that the other conditions
with are obviously satisfied, we have proved the following classical result of I. BABU ŠKA [Bab73]:
0
Theorem 2.2 The problem (2.3) possesses a unique solution. Furthermore, there exists a constant
independent of and but depending on with
" 1" ( " ?" % 0 L " " ( "2" O Q D F U #
14
Imposing Dirichlet conditions in a weak sense
Our next Corollary shows that with a minor additional assumption represents the normal flux of
on the boundary. This is very important in many domain decomposition algorithms, where the
Neumann values on the subdomain boundaries are interchanged.
, where Corollary 2.1 Assuming additionally Lagrange multiplier can be represented by
for denotes the solution of (2.3), the
(cf. [Bab73] ).
Proof: Integrating by parts the first equation of the weak formulation (2.3) yields
S $#
we observe, that If we restrict the space of test functions in (2.7) to 0
distributional sense. Therefore there holds
hence . Additionally is sufficiently regular because of Theorem B.11.
(2.7)
holds in the
2.2 Discretization and stabilization
6/ ( % ")" ( (2.8)
(cf. Lemma
Furthermore, according to Lemma B.3 there exists a quasi-interpolation operator
B.1).
and a constant 0 with ( % 0 ( !#" $ &%
(2.9)
( % 0 ' Q ( !)( (2.10)
(
(
!#"
%
" 2" 0 ")*"
(2.11)
, % $ % % % , , . Here is a face resp. edge of an element
for , % *$!+
+ (cf. Lemma B.3),
/. which have at least one corner in common
- is the union of all
(
with , and - is given by - 0 21 435 - .
In a next step the equation (2.3) is discretized. To this end we consider conforming approximations and . For we use a finite element
discretization, whereas
the choice of the space is arbitrary, but should satisfy . Given an admissiinto simplicial elements the space is given by
ble, shape-regular decomposition of
, . denotes the diameter of an element
. A detailed description of the definitions can be found in section B.6. For functions
the following standard inverse inequality holds
B
B
7
2.2 Discretization and stabilization
15
Next we analyze the corresponding discrete problem:
5
#
Find
8
8
"
(2.12)
Again Theorem 2.1 yields
be elliptic on 4 . If the discrete spaces
(2.13)
0 5 5 5 5 " " ") " 0 Lemma 2.2 Let
and satisfy
D 8
"
,5
7 C;=
>
B
7 - ; =
?A
>
then the discrete problem (2.12) possesses a unique solution.
Remark 2.1 Using more of the theory of linear mixed problems and assuming (2.13) we can also
derive the following error estimate
" ( " " % 0 5 5 " " ( 5 5 " " (2.14)
is independent of the discretization spaces but depends again on where the constant 0 "
,5
7 -
B
6
,5
7 (cf. [GR86]).
Considering the last Lemma, we recognize two typical problems. First the constraint (2.13) is in
general not satisfied. The inf-sup condition (2.13) means that the space has to be sufficient
’rich’ compared to the space of the Lagrange multipliers / . The problem of construction of finite
element functions satisfying (2.13) is discussed in detail by J. P ITK ÄRANTA [Pit80, Pit79]. For
satisfies (2.13). This choice corresponds to the standard
example the simple choice method, where the boundary conditions are enforced strongly.
The second problem is the dependence of the constant in (2.14) on . In general, it will increase,
if tends to zero. So we loose control of the error in the advection dominated case. This case is
given, when the local Peclet number
" N" D A C D F FHT ,
is large. And, in fact, using a standard discretization it is well known that there may arise spurious
oscillations of the computed solution (cf. [RST96]). Both problems can be solved by adding
stabilization terms to our variational formulation (2.12).
In order to circumvent the inf-sup condition (2.13) we add a term to the second line of (2.12),
3
namely
3
;
7
35
D
(
(2.15)
with
,
. is the mesh on the boundary induced by the mesh restricted
to the boundary . This idea is not new. For instance, similar but symmetric terms are also used
in [BH92, Ste95]. Alternatively the condition (2.13) can be ensured by enriching the space 2 by
bubble functions (cf. [BFMR98, BM01]).
16
Imposing Dirichlet conditions in a weak sense
In the interior of the domain the standard streamline diffusion method (SUPG-method), introduced by T.J.R. H UGHES and A.N. B ROOKS [HB79], is used in order to damp the oscillations
in streamwise direction. SUPG stands for Streamline Upwinding Petrov/ Galerkin. Therefore the
and the linear form are replaced by
bilinear form
4 ; 43 5
;
4 35 #
For the parameter
we adopt the proposal of [RST96]
and
define
% " N" A C D F in the
and advection dominated regime for
for
.
7
7
Remark 2.2 Of course there are many other methods to stabilize the advection diffusion equations
by adding additional terms in the interior. A unified presentation is given by R. H ANGLEITER , G.
L UBE [HL98].
Another important approach, which is becoming quite popular, enriches the space of test functions
with a space of bubble functions. It can be shown that it is equivalent to the SUPG method in the
case of constant coefficients and linear elements (cf. [BBF93, BMS00, BFHR97]). A common
general framework for all approaches is given by the theory of subgrid scale models [Hug95].
There are also attempts to damp the oscillations in crosswind direction. But inserting additional
terms in a consistent way requires a nonlinear variational formulation for higher order finite
element spaces (cf. R. C ODINA [Cod93], T. K NOPP, G. L UBE, G. R APIN [KLR01], [KLR02]).
; 35 The oscillations on the boundary
are controlled by adding
+
+
% ? (2.16)
is equal to on the
to the first equation
of (2.12). Here of the boundary and zero elsewhere. Adding
inflow part 7
the corresponding term is essential for discretizations of hyperbolic problems (cf. [JP86]). We
have been motivated to use the term (2.16) by some recent papers about discontinuous Galerkin
methods (cf. [Fre01, BO99, HSS02]). After adding all these terms we get the following stabilized
variational formulation: Find such that
; 35 +
7
8 ;
3
7
35
%
D
3
+
+
8
(2.17)
(2.18)
3
, # . At the moment the choice of the parameters
for all ,
has not been fixed.
We have only imposed
,
. They will be determined later on with the help of
2.2 Discretization and stabilization
17
λ
λ
19
x 10
1.5
3
1
2
0.5
1
0
0
−0.5
−1
−1
−2
−1.5
1
−3
1
0.8
0.8
1
0.6
0.6
0.8
0.6
0.4
0.4
0.2
0.4
0.2
0.2
0
(a)
1
0.8
0.6
0.4
0.2
0
0
, , (b)
0
, , Figure 2.1: Approximation of the normal derivative of (2.19) on the boundary in the diffusion dominated
case: On the left hand side the discrete inf-sup condition (2.13) is fulfilled. In Figure 2.1 (b) the condition
is not satisfied.
+
3
the a priori estimate. But already now we make the following observation: If we assume
for , then our scheme (2.17), (2.18) reduces in the hyperbolic limit (
4 %
;
%
743 5
;
7
3
35
D
(
#
) to ,
Now the two equations are decoupled. The first equation is a stabilized scheme for the hyperbolic
problem, where the boundary condition on the inflow is imposed in a weak sense (cf. [JP86]).
Having computed , we can find the Lagrange multiplier part by the second line. So we can
control the multiplier even in the hyperbolic case. Of course the computation of the Lagrange
multiplier part makes only sense, if the continuous solution of the hyperbolic limit problem is
sufficiently regular.
+
3
for Remark 2.3 If
,
is chosen in the advection dominated regime, the
solution of the reduced problem is approximated
and not the solution of the elliptic
+ on the outflow
3
case.
If we choose the stabilization parameters by
,
, for we
approximate for all the elliptic case, because for small the additional stabilization term in
(2.18) can be neglected and the equation reduces to
!
8 #
"
#
With the help of the a priori analysis suitable choices will be given for both parameter strategies.
18
Imposing Dirichlet conditions in a weak sense
λ
λ
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
1
−1.5
1
0.8
1
0.6
0.8
0.6
0.4
0.2
0
0
0.4
0.2
0.2
0
0.8
0.6
0.4
0.4
0.2
(a)
0.8
1
0.6
, , (without stabilization)
(b)
0
, , (with stabilization)
Figure 2.2: Approximation of the normal derivative of (2.19) on the boundary in the convection dominated
case: In both cases the condition (2.13) is satisfied.
In order to clarify the further proceeding and to visualize the described problems the following
example is considered:
Example 2.1 Let the right hand side and the boundary condition be chosen in such a way that
becomes the exact solution of
(2.19)
?
in
on
#
Although the solution is smooth, the normal derivative possesses jumps at the corners. For the
discrete spaces we use quasi-uniform meshes with mesh sizes resp. and linear elements.
In Figure 2.1 (a) we see an approximation for the normal derivative, where we have used the
SUPG stabilization in the interior but no boundary stabilization. The choice of the discrete function spaces satisfies the Babuška-Brezzi condition (2.13). Choosing a finer boundary mesh the
condition (2.13) is violated and indeed in Figure 2.1 (b) we observe a poor approximation. Next
we consider the convection dominated case. Even though the constraint (2.13) is satisfied, we get
poor results (cf. Figure 2.2 (a)). But with stabilization on the boundary we can control the normal
derivative (cf. Figure 2.2 (b)). (The stabilization parameters are given by (2.34), (2.35).)
So the figures show very clearly the positive effect of the stabilization.
2.3 Stability
The issue of this section is the proof of the stability of the discrete scheme (2.17), (2.18). By simply
adding (2.17) to (2.18) we obtain the following equivalent formulation: Search for such that
(2.20)
4 4 4 4
2.3 Stability
19
4 4 ; 3 5 4 %
; 4 4 35
;
4 % #
4 4
35 with
+
7
8
3
8
7
+
7
For the stability estimate we need two well known results. First we need a stability estimate of the
streamline diffusion scheme in the mesh dependent norm
" " " " ; 43 5 " " ( #
Taking into account that the functions do not vanish on the boundary, according to
G. L
( [Lub94], Lemma 2.1), we get the estimate
; % " " % # (2.21)
35 7
UBE
7
Furthermore, we need a second inverse inequality (cf. [Ste95], Lemma 3):
0 , not depending on the mesh size, such that
% 0 Q ( Lemma 2.3 There exists a constant
for all
, + .
(2.22)
Now we are prepared to prove the 3 main result of the section:
,3 satisfy
%
'0
Theorem 2.3 Let the parameters
3
3
D
B (2.23)
0 , , and are defined in
(2.24)
4 " " ( (
(
is true for all # denotes the norm " " . Here ""
" " " "
with
")2" " *" % ; 35 ")2" ( (
;
" " 35 E" " . Therefore the scheme (2.17), (2.18) possesses a unique solution.
for , , where is the space dimension. The constants
for (2.22), (2.8) and (Ass. 2a). Then the inequality
D
D
D
D
D
3
7
+
3
B
7
20
Imposing Dirichlet conditions in a weak sense
4 using (2.21)
4 4 4 " " 4 % ; 3 5 ") " ; 4 4 #
35
, +
we arrive at
Then, using the Young inequality and Lemma 2.3 for
;
&
; 4 4 " 4 " E" 4" " " 35
35
" 4" ; 3 5 E" " ( #
;
" 4" '0 D F
Proof: Simple computation shows for
+
7
3
7
3
(2.25)
3
7
3
7
7
7
3
3 5
Using
the inverse inequality3 (2.8) and the restriction (2.23) we obtain
3
0 ( D 0 B F%
hence the assertion, because
has at most
" 4G" D F % faces resp. edges.
3
" D F "
Let us take a close look at the estimate (2.24). We observe that, in contrast to the SUPG-method
with homogenous Dirichlet conditions we
have some additional control at the boundary. This will
3
be used for the a priori estimate.
has to be small because of the restriction (2.23) and
But on the other hand the parameter
therefore the norm of the Lagrange multiplier part gives not very much control. Taking into
account that typically there are strong boundary layers on the outflow, we cannot expect a stronger
norm, because the Lagrange multiplier part is given by the normal derivative of the solution .
" "
2.4 A priori analysis
+
3
The aim of this section is the derivation of an a priori result. Then, this estimate will be used to
provide the parameters
. For the proof we need the following continuity estimate of the
streamline diffusion part, which can be proved analogously to G. L UBE ([Lub94], Lemma 2.2)
and which holds for all
.
0
0 %
0 " " 0 0 ;
( !#"
(
L
U
M" R"
3 5
% 7
(2.26)
where is a constant depending on the reaction coefficient and the interpolation estimate (2.9).
This inequality enables us to prove the following continuity estimate.
2.4 A priori analysis
21
0 4 % 0 ")" " 4" ; " " 0 35
(
0
;
" E" ( ! (
0 35
0 ; K" N" ( ( !#" (2.27)
0
" Lemma 2.4 Assume
, , ,
and 4 # . Furthermore,
with the help of the quasi-interpolation operator . Then for all
define
3
+
3
3
7 3
+
7
7 35
is true.
; 35 ; 35
Proof: We start again with our bilinear form
4
+
7
7
3
4 %
4
4
#
The bilinear form will be estimated term by term. First using the Cauchy-Schwarz and the Young
inequality we have
4 ;
% 0 " 4"
for all
0 Q
% ; E"4" "4" E" " 7 35
3 7 3 5
;
% 0 "4" " " 0 7 35
%
3
00
;
7
3
35
;
3
35
7
Q
" " (! (
by the interpolation estimate (2.10). Similarly we obtain
% 0 ; 3 5 E")4" 0 ; 3 5 " " #
we can estimate the first row of our bilinear form by
Defining (2.26):
; %
35 % 0 " " 0 ; L " N" ( U ( !"
0 35
; %
35 +
8
7
+
7
+
7
7
7
+
22
Imposing Dirichlet conditions in a weak sense
;
%
35 ")4" 0 %
#
( !#"
;
;
(
0
L
U
0 35 M" R" 0 35 E" " +
% 0 " 4 " 7
+
7
7
; %
35 % 0 )" " 0 ; L M" R" ( U ( !#"
0 35
(
0
;
L
0 35 " " Now the interpolation estimate
(2.10) yields
+
7
3
7
+
7
U ( !)( #
In order to estimate the last missing term, we use the Cauchy-Schwarz inequality and (2.10) again
to compute
;
7
3
35
Q
;
%
4 % " 4" 35
% 0 " 4" " " 0
0 % 0 " 4" " ?" 0 ; ( ! (
0
0
35
3
7
3
% ;
where we have employed
3
35
7
(
% 0 ;
35
3
7
7
(! ( #
Now collecting the estimates for all terms, we obtain the assertion.
%
(2.28)
Combining the last Lemma 2.4 and the stability result of Theorem 2.3 we get
(
" " -
% , be the solution of the continuous
Theorem 2.4 Let ,
problem (2.3) and be the discrete solution of (2.17), (2.18). Furthermore, let
the stability condition (2.23) be satisfied. Then
the error
can be estimated by
3
+
6 5 7 5 5 ; " 7 35
7
" ( !#"
;
(
" N" 35
; M" E" ( 35
7
3
+
3
(! ( #
(2.29)
2.4 A priori analysis
23
Proof: Suppose . Defining
D
with the help of the quasi-interpolation operator
the Galerkin orthogonality
(2.30)
(2.31)
, and using (2.17), (2.18), and (2.3), we obtain
and therefore by Theorem 2.3 and Lemma 2.4:
"
(
" %
% 0 " " ( ; "
0 35
(
0 ; M" E" 0 35
0 ; K" N" ( ( !" #
0 35
3
3
+
7
3
+
7
3
7
" (2.32)
(! (
Next we estimate the -part with the help of the interpolation estimates (2.9), (2.10):
%
" 1" " " % ; 3 5 E" '" % 0 ; " N" ( ( !#" 0 ; K" E" ( ( ! (
43 5
35
The constant 0 depends again on in the
where we have used the definition of (cf. [RST96]).
interior of the domain . Now we choose 0 in (2.32). Thus we arrive at the assertion by
(
( %
(
(
" ; " " " " " ; " "
- " " K" N" ( ( !"
35
35
;
(
( !)( #
"
E
"
35
+
+
7
7
7
3
+
7
3
+
7
3
7
+
3
We are now in a position to derive a reasonable choice for the parameters
by minimizing
the right hand side of the a priori estimate (2.29) and taking into account the condition (2.23).
3
+
Starting with
645 ,7 5 5 ; 7 35
"
" 24
Imposing Dirichlet conditions in a weak sense
+
;
35
7
In order to equilibrate +
3
+
. Then we
consider
+
3
3
and
3
3
, we propose
for
& +
+
(2.34)
for (2.33)
& . The second parameter is then determined by
(! ( #
3
for a suitable global
M" E" ( 7
we observe that we have to3 choose
3
#
+
+
3
(2.35)
This choice also satisfies (2.23) and, as requested in section 2.2, we can control the Lagrange
,
for .
multiplier part even in the hyperbolic limit, because there holds
Remark 2.4 Using our proposed choices (2.34), (2.35) the a priori estimate simplifies to
" " ( - ; 3 5 " N" ( ( !#"
;
5 5 " " 35
(! ( #
;
(
"
E" 3
5
7
6
5
7 7
7
Assume that the discrete Lagrange multiplier space is defined by
Q (2.36)
then we obtain under the assumptions of the previous Theorem
6 5 7 5 5 ; " 7 35
; " 35
7
( ! ( #
Thus the error estimate is of the same order as in the case of the standard SUPG-method with
homogenous boundary data (cf. [RST96], [Zho97]).
Remark 2.5 If we want to approximate the elliptic case on the outflow, cf. Remark 2.3, the stabilization parameters can be chosen by
+
+
3
3
(
(2.37)
!
!
#
With this choice the a priori estimate possesses the same convergence order as given by (2.34),
(2.35).
2.5 A posteriori analysis
25
2.5 A posteriori analysis
Now we derive an a posteriori error estimate. Hence it is possible to control the error by adaptivity. Our approach is based on the work of S. B ERRONE [Ber01, Ber99]. He derives sharp error
estimates for the Oseen equations. We adopt his method to our stabilized scheme.
But first we need some estimates of the quasi-interpolation operator of (2.9) -(2.11) in the norm
, +
, which is the energy norm restricted to (cf. Lemma B.4).
" " " "
0 ") *" ") 2" ( " " ( and
for - , ,
Q Q Lemma 2.5 There exists a constant
Defining the error by
of (2.3) and .
Estimate of such that
% 0 ") " !#"
% 0 I Q Q " " ! (
% 0 " " !#"
% 0 " " ! "
,
or
is a face of a
+
#
and , where is the solution
is defined by (2.17), (2.18), we start our analysis with the part
#
The first equation of our variational formulation (2.3) yields for
(2.38)
Choosing we add to (2.38) the first line (2.17) of our stabilized scheme with test function
and obtain
; % 35 +
7
or
; % ; 43 5
35 7
+
7
#
26
Imposing Dirichlet conditions in a weak sense
Splitting one of the integrals over
applying integration by parts yields:
; L 35
; 35 7 +
into the sum of integrals over the finite elements
7
;
1 ;
%
7 35
and
U
#
Now we split the set consisting of all faces resp. edges of the decomposition into the set
and the complement . Furthermore, we denote by the jump on a face
. The
sign of the jump on
is not important, because we are only interested in the absolute
value of the jump. Then we obtain
%
; L U
35
; ; " " " " 5 35 35
; % #
35 7
7
+
7
C;
7
###
% ; %
35 !# "
;
L
" " " "
0
3 5 ! "
F
" N" A C D " " " " U
! (
;
L
" " " " U
Q
3
5
! (
;
L
" " " " U Q
5
3
5
Q Q , if + or is a face resp. edge of some , .
where we have used
The constant 0 is given by the interpolation estimates of Lemma 2.5. Since the mesh is shape
regular, there exists a constant 35 such that
; " " ! " % 35 " " ; " " ! ( % 35 " " #
(2.39)
35
35
Then using
" " %
Next we use the Cauchy-Schwarz and the Young inequality and the properties of the interpolation
, operator (cf. Lemma 2.5). Then we get for arbitrary
+
7
7
7
7
7
C;
7
2.5 A posteriori analysis
27
5 we obtain
;
%
% " " 3 5 ;
;
F
" " 0 M" R" ACED " " 35
5 3 5
; " " (2.40)
35
Next we analyze the additional stabilization term. Using
with a constant 0
. we
get
; %
3 5 ;
%
35 #
; %
3
5
Applying the same technique as above we can deduce with any constant % ;
L
U
F
Q " " ACED " *" 35 0 ; " " !) ( ; % #
35
35 and setting the constants by +
7
7
7
C;
7
+
7
+
7
+
7
+
7
+
7
7
constant given by the interpolation estimates. Choosing 5 0 is the and,
using
% " " " *" Q F O D
Again,
such that
yields the following estimate for :
Lemma 2.6 For any
+
there holds
;
" " 35 ; " N" A C
0 43 5
; " " % % " " " *" O Q D F
D F " " ; 5 3 5 " " ; L M" " A C D F U " *" (2.41)
Q
35
35 .
with a suitable constant 0 which is independent of and the diameter of triangles 7
7
7
7
+
7
;
28
Imposing Dirichlet conditions in a weak sense
This estimate is not a real a posteriori estimate, because the unknown error of the Lagrange mul
tiplier stands on the right hand side. In a next step an estimate for is derived with the error
on the right hand side. Combining both estimates will give us the desired estimate.
Estimate of
In order to derive an upper bound for the error of the Lagrange multipliers, we start with the
continuous Babuška-Brezzi condition proved in Lemma 2.1:
0 6 5
7 ; =
>
BC7
ORQD F
?A
;=
0 #
(
")2" " "
>
Now inserting (2.38) and adding the discrete formulation (2.17) with the test function
yields for the error of the Lagrange multipliers
0 " " %
O QBD F ")*" ( ; % #
35 yields
Integration by parts over the triangles ; %
0 " " %
(
O Q D F ")*" 35 ; L U
43 5
; ; " " E") *" #
5 3 5 35
BC7
?A
;=
>
+
7
+
BC7
;=
?A
7
>
7
7
7
2;
Applying the interpolation estimates (2.9), (2.10), (2.11) we obtain by the Cauchy-Schwarz inequality
0 " "
O Q D F ")*" ( ; E" " U Q L ; ( ! ( U Q
L
(
O Q D F ")*" 3 5 5 35
; " " U Q L ; " N" A C D F " " U Q L ; ")*" ( !#" U Q
L
35
35
35
Q
;
; ")2" ( !" U Q
L " " A C D F " *" U L
35
35
;
" U Q L ; ( ! ( U Q #
L " 35
35
BC7
BC7
;=
?A
?A
7
;=
>
>
7 :;
7
7
C;
7
+
7
7
7
7
2.5 A posteriori analysis
29
Proceeding as in the estimate of we get (cf. (2.39))
;
0 " "
O Q D F )" 2" ( 5 3 5 E" " ; " N" A C D F " " ; " " 43 5
435
Q #
;
L
F
" " " " A C D " 2" U 3
5
BC7
7
?A
7
;
7
+
7
O Q D F ")*" ( The first term of the right hand side can be estimated with the help of integration by parts:
( ( (
F
O Q D ")*" ( % - ( Q " R" ACED F " " ( " N" A CED F " " ( Q Q
O Q D F ")2" ( % % - " N" A CED F M" R" ACED F " @" ACED F " "B
Q #
Q
F
" R" A C D
%
% . This implies
In the last step we used the trace theorem and Lemma 2.7 With a suitable constant 0 , there holds the following estimate for 0 " " % " R" ACED F " N" A C D F " @" A CED F " " ; E" " " " ACED F " " A D F 5 3 5
; " R" ACED F " " ; " " 43 5
435
#
;
L
F
"
"
" " A C D " *" U
3
5
B27
?A
;=
B27
>
;=
?A
BC7
?A
>
;=
>
:
7
7
7
7
C;
+
Unfortunately the upper bound of the error of the Lagrange multipliers also includes an unknown
variable, the error , on the right hand side.
Combination of both upper bounds
Now we combine the results of Lemma 2.6 and Lemma 2.7. This is possible because there is one
remaining degree of freedom, the constant in the estimate of Lemma 2.6. Now we choose
0
" R"BA C D F M" N"BA CED F M
" R"BACED F " @"BACED F
30
Imposing Dirichlet conditions in a weak sense
0
with the constant
of Lemma 2.7. Then, inserting the estimate of Lemma 2.7 into the upper
bound of Lemma 2.6 yields the following a posteriori estimate for the error 2 .
Theorem
2.5 Let us assume, that is the solution of (2.17), (2.18). Furthermore, let
be the corresponding continuous solution of (2.3). Then there holds
" " - " " ACED F " N" A C
;
F
" N" A CED
7
" " 35
; L U L " Q
35
7
where
;
7 35
D F M" N" A C D F M" " A C D F " *" O Q D F
;
" " " " 5 35
+
7
is defined by
" A C D F U " *" Q Q , with ,
C;
or
.
In order to derive the estimate (2.42) we have used the obvious inequality (2.42)
.
Remark 2.6 The error estimate (2.42) contains the same terms in the interior of the domain
as the upper bound derived by R. V ERF ÜRTH [Ver98], Prop. 4.1, or by G. S ANGALLI [San01],
Theorem 2.
Remark 2.7 Unfortunately the term
"
*" O Q D F in the first line of (2.42) depends on the global
behavior of . So the estimate is not well adapted for an error indicator used in an adaptive
implementation. But for linear elements in the interior the technique of inverse inequalities given
by [DFG 01] can be used. If # is the nodal interpolant corresponding to the boundary mesh of a sufficiently smooth function , we obtain
" *" O Q D F - " ;
" Q F " 2" O Q D F O D
(
#
" " " *" Q F
(2.43)
O
D
35
Neglecting the data error ") *" Q F and inserting (2.43) the error estimate (2.42) can now
O D
be applied to an adaptive scheme of (2.17), (2.18).
A more direct approach is given by B. F
[Fae00]. She shows for boundary integral
Q
methods, how the residuum in the -norm can be localized directly without using inverse
-
7
AERMANN
inequalities.
3
3
Remark 2.8 We have only used (2.17) for the a posteriori estimate. But the choice of the param in (2.18)
eters
implicitly has some influence on the estimate. For
larger the weighting of
( 5
becomes stronger than the weighting of and vice versa. So for large
"
"
"
2" 2.6 Numerical results
31
mesh for λ
mesh for u
1
common mesh
1
1
0.9
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0
(a) Mesh for 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
(b) Mesh for 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c) Common mesh
Figure 2.3: Finite Element meshes for the interior part (a), the Lagrange multiplier (b) and the common
mesh (c).
3
"
3
*" we loose control of the second term and therefore might be large. An appropriate
choice of
has been derived with the help of the a priori analysis.
Remark 2.9 Of course, now we can simply obtain an a posteriori estimate for the error of the
Lagrange multipliers , too, if we insert (2.42) into the estimate of Lemma 2.7.
2.6 Numerical results
All the computations are made with MATLAB
6.5 using routines of FEMLAB
2.3. For
. Moreover simplicity all numerical experiments are restricted to the unit square
is given by the set of continuous, piecewise linear functions. Similarly consists of continuous,
piecewise linear functions. Therefore we need a grid for in (cf. Figure 2.3 (a), )
and a boundary mesh for (cf. Figure 2.3 (b), ). Although our analysis is still valid
for non quasi-uniform meshes, in our numerical experiments both meshes are quasi-uniform with
which
mesh sizes resp. . In our algorithm we have to compute boundary integrals on
contain functions of and . To this end we introduce a third mesh given by the union of
the mesh points of the other meshes (cf. Figure 2.3 (c)). This procedure only works well in two
dimensions. In 3D the construction of a common refinement is too costly. But the problem can be
circumvented by using suitable quadrature formulas. For some experiments compare Y. M ADAY,
F. R APETTI , B. W OHLMUTH [MRW02].
#
A first smooth example
+
#
3
Let us consider again the first example with , . For all computations we have
chosen
, (except in Figures 2.9, 2.10). Moreover, we concentrate on the hyperbolic
choice (2.34), (2.35). But the elliptic choice (2.37) gives quite similar results.
Example 2.1 Let be a solution of
in
on 32
Imposing Dirichlet conditions in a weak sense
λ
error of the Lagrange multiplier
0
10
1.5
1
L (Γ)
0.5
2
0
−1
10
−0.5
−1
ε =1
ε =0.1
ε =0.0001
−1.5
1
0.8
1
0.6
0.8
0.4
(a)
0
−2
0.6
10
0.4
0.2
−3
−2
10
0.2
10
"
0
(b)
of (2.44)
0
10
" A D F for Figure 2.4: In (a) the normal derivative of (2.44) is plotted ( , -error of the Lagrange multiplier part in dependency on the mesh size ( where the right hand side
−1
10
hint
, ). The
) is given in (b).
and the boundary condition are chosen in such a way that
(2.44)
?
becomes the exact solution.
#
#
Although the solution is smooth, theflux
has some
discontinuities
because of the corners of the
domain (cf. Figure 2.4 (a), ,
, ). The peaks of the flux at the
corners are caused by the continuity of the test functions.
#
In a next step we verify the convergence rates numerically. Therefore we use the same problem
and
as above.
Three different values of the diffusion parameter are investigated ( , ). In order to avoid matching grids we have chosen . We obtain the following
results, which are plotted in Figure 2.4 (b) and Figure 2.5 (a), (b).
For the solution component we observe a convergence rate of order with respect to the
norm. The convergence
rate for
the energy norm is of order in the convection dominated case for and order
for . For the Lagrange multiplier part we obtain a rate of about . This seems to
be very small. But the convergence of the Lagrange multiplier part is influenced by two facts. On
the one hand the convergence behavior
is caused
by
the definition of the norm. The norm is
very weak because of . On the other hand there holds the
following approximation estimate
#
# " "
" " #
" ,5
645 7 5
" " #
" &"BA D F
(
K( #
"
" 1" #
" " In summary one can say that the convergence is better than expected. The theory predicted for the
energy norm convergence of order in the diffusion dominated case and of order in the
convection dominated case. For the Lagrange multiplier part we have expected a rate of in the
strongly diffusion dominated case.
2.6 Numerical results
33
error in the interior
−1
error in the interior
−1
10
10
−2
10
−2
10
−3
2
L (Ω)
energy
10
−4
−3
10
10
−4
10
ε =1
ε =0.1
ε =0.0001
−5
10
−6
10
ε =1
ε =0.1
ε =0.0001
−5
−3
−2
10
10
(a)
"
" A D F
−1
hint
for 10
0
10
10
−3
10
(b) " −2
10
"B
−1
0
10
h
int
for 10
Figure 2.5: The -error of the interior part (a) and the error in the energy norm of the interior part (b) in
dependence on the mesh size. ( )
#
In the next figure we have used the same parameters as above. But this time we have fixed the
mesh in the interior ( ). Then we consider what happens, if the boundary mesh is refined.
The results are plotted in Figure 2.6 (a), (b), and 2.7 (a). We observe the expected behavior. It
error in the interior
−2
error in the interior
−1
10
10
−2
energy
2
L (Ω)
10
−3
10
−3
10
ε =1
ε =0.1
ε =0.0001
−4
10
−3
−2
10
10
(a)
"
−1
10
"BA D F
h
λ
ε =1
ε =0.1
ε =0.0001
−4
0
10
10
−3
10
−2
10
(b) " h
λ
"B
Figure 2.6: The error for different boundary meshes and fixed interior mesh. ( −1
0
10
10
)
does not make sense to use a much finer boundary mesh than the interior mesh. In this example
maybe the situation differs if we have very rough data on the boundary or the solution is not
very smooth.
34
Imposing Dirichlet conditions in a weak sense
error of the Lagrange multiplier
0
error of the Lagrange multiplier
10
−0.2
10
−0.3
10
−0.4
L2(Γ)
2
L (Γ)
10
−1
10
−0.5
10
−0.6
10
ε =1
ε =0.1
ε =0.0001
−2
10
−3
−2
10
−1
10
(a)
"
0
10
−3
10
" A D F
hλ
ε =1
ε =0.1
ε =0.0001
−0.7
10
−2
10
−1
10
"
(b)
0
10
10
" A D F
hint
Figure 2.7: In (a) the error is plotted in dependence on the boundary mesh size for fixed interior mesh.
) In (b) the boundary mesh is fixed. ( )
( error in the interior
−1
error in the interior
−1
10
10
−2
10
−2
10
−3
2
L (Ω)
energy
10
−4
−3
10
10
−4
10
−5
10
ε =1
ε =0.1
ε =0.0001
−6
10
−3
−2
10
10
(a)
"
−1
10
" A D F
h
ε =1
ε =0.1
ε =0.0001
−5
10
0
10
−3
10
−2
10
int
(b)
" h
int
"
−1
10
0
10
Figure 2.8: The error is plotted in dependence on the interior mesh size. The boundary mesh is fixed.
)
( #
Vice versa we fix the boundary mesh ( ) and vary the mesh size of the interior mesh.
The results are printed in Figure 2.7 (b) and Figure 2.8. Again we observe, at least for the error of
the Lagrange multiplier part and the energy error in the interior, that in this example refining the
interior mesh much stronger than the boundary mesh does not yield 3 better results.
+
3
#
And finally we test the robustness of our choice of the parameters
and
. We start varying
the parameter
for fixed meshes ( ). The results can be seen in the plots
, 2.6 Numerical results
35
error of the Lagrange multiplier
1
error in the interior
−3
10
10
0
2
2
L (Γ)
L (Ω)
10
−4
10
−1
10
ε =1
ε =0.1
ε =0.0001
−2
10 −3
10
−2
−1
10
0
10
"
"BA D F
β
10
−5
10 −3
10
2
10
−2
−1
10
(a)
10
1
ε =1
ε =0.1
ε =0.0001
0
10
(b)
"
β
10
1
2
10
10
" A D F
Figure 2.9: The dependence of the error on the constant . The plots show the robustness of our choice of
, , )
the stabilization parameter . ( error of the Lagrange multiplier
0
L (Ω)
−1
2
2
L (Γ)
10
10
ε =1
ε =0.1
ε =0.0001
−2
10 −3
10
error in the interior
−3
10
−2
−1
10
0
10
(a)
"
α
10
" A D F
1
10
−4
10
ε =1
ε =0.1
ε =0.0001
−5
2
10
10 −3
10
−2
−1
10
(b)
0
10
"
α
10
1
2
10
10
" A D F
Figure 2.10: The error in dependence on the constant . The plots show the robustness of our choice of
, , )
the stabilization parameter . ( 3
3
of Figure 2.9 (a), (b). We observe that it is very important, that we do not choose the parameter
too small, because then we loose control of the Lagrange multiplier part. If we choose
very
large, we cannot improve the error of the Lagrange multiplier part very much, but we observe an
+
increase of the -error of .
Then we performed the same computations for the parameter . The results are plotted in Figure
36
Imposing Dirichlet conditions in a weak sense
+
+
2.10 (a), (b). We observe that the choice of
influences the approximation error only in the
+
diffusion dominated case. And there we see, that a larger value of
improves the error of the
interior solution. For a very small value of
the interior error deteriorates. For the error of the
Lagrange multiplier part we have the opposite behavior.
An example with boundary layers
In the first example the exact solution was smooth and independent of . In the next example
we investigate the case of boundary layers at the outflow. Our test case is also discussed by G.
M ATTHIES , L. T OBISKA [MT01] in the context of nonconforming finite element discretizations.
Example 2.2 We consider the problem
The right hand side
in
on
#
and the Dirichlet boundary condition are chosen such that
becomes the exact solution.
Again the boundary mesh is chosen three times finer than the mesh in the interior ( ).
In Figure 2.12 (a) the convergence
of the method in the
norm is plotted. As expected the
convergence rate is about in the singularly perturbed case.
We cannot expect more, since even
the approximation error only converges with a rate of . This behavior, however, is only local.
Away from the layers we observe standard convergence rates.
Compare Figures 2.11 (b), 2.12 (b),
.
where we consider the error in the domain
.
There is one problem, that should be noticed. In the convection dominated case the approximation
of the Lagrange multiplier
3 part is poor, because, as we have shown in the first part, we approximate
for the flux of the hyperbolic limit problem. If we want to approximate the real flux, the
stabilization parameters
,
, have to be chosen smaller. Then it is possible to get better
approximations even in the boundary layer region. For example the elliptic choice (2.37) gives
better results.
# An example with non-uniform meshes
The last example is given by S. B ERTOLUZZA (cf. [Ber03b]). She tries to accelerate the poor
convergence of the Lagrange multiplier part by choosing non-uniformly refined meshes for close to the boundary like the one in Figure 2.13 (a). The Lagrange multiplier space is chosen as
the restriction of the interior space to the boundary. We study the case of non-uniform meshes by
using the example of S. B ERTOLUZZA:
V
Example 2.3 Let the right hand side
?
?
and the boundary conditions be chosen, such that
becomes
the exact solution of in .
2.6 Numerical results
37
error in the interior
−1
10
−2
10
0.7
−3
10
energy
0.6
0.5
0.4
−4
10
0.3
0.2
0
0.1
−5
10
0.2
ε =1
ε =0.1
ε =0.0001
0.4
0
0.6
−0.1
−6
10
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) The solution of Example 2.2 for
−3
−2
10
10
1
#
(b)
" −1
0
10
hint
10
" Figure 2.11: In (a) we observe the sharp layers of the solution of Example 2.2. In (b) the dependence of
the error away from the layers on the mesh size is plotted.
error in the interior
0
error in the interior
−2
10
10
−1
10
−3
10
−2
10
−4
L (Ω)
−3
10
2
2
L (Ω)
10
−5
10
−4
10
−6
−5
−6
10
10
ε =1
ε =0.1
ε =0.0001
10
−3
−2
10
−1
10
h
10
ε =1
ε =0.1
ε =0.0001
−7
0
10
10
−3
−2
10
−1
10
h
int
(a)
"
" A D F
10
0
10
int
(b)
"
" A D F
Figure 2.12: The error of the
example: On the left hand side (a) the error in the whole domain is
second
plotted. In (b) the error in
is plotted.
38
Imposing Dirichlet conditions in a weak sense
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) (a) non-uniformly refined mesh
Figure 2.13: In (a) a non-uniformly refined mesh is plotted. And in (b) the Lagrange multiplier of Example
, )
2.3 is printed. ( By this approach she obtains for the approximation of
the Lagrange multiplier with degrees
of freedom an
-error amounting to about
. With almost the same number of d.o.f.’s
our method also yields an error of about . However, we use a uniform mesh in the interior
(d.o.f.’s for : 9036, d.o.f.’s for : 2856) and a much finer mesh on the boundary. If we use the
(d.o.f.’s for : 6969, d.o.f.’s
non-uniform mesh of Figure 2.13 (a) the error can be reduced to
for : 4000). This illustrates the advantage of our approach to use arbitrary meshes.
#
2.7 Conclusions
In this chapter we have proposed a stabilized scheme for the advection diffusion equations. We
proved that the scheme is stable for arbitrary finite element discretizations. Note, that the finite
element discretization need not to be quasi-uniform and for the Lagrange multiplier space any
discretization can be chosen. Moreover, compared to the standard streamline diffusion method
with homogenous boundary conditions, optimal a priori and a posteriori estimates are derived.
With the help of the a priori estimate it was possible to derive a suitable choice of the stabilization
parameters. The numerical experiments confirm the predicted convergence rates. We have also
shown that the algorithm is robust with respect to the choice of the stabilization parameters. Additionally we give a second choice which resolves the original outflow boundary conditions. To our
knowledge detailed numerical experiments for methods where Dirichlet boundary conditions are
weakly worked in have not been carried out in the literature so far.
The residual based a posteriori estimate enables us to create an adaptive algorithm with refinement
in the interior and on the boundary. It is the first a posteriori error estimation for this kind of
methods. But an implementation will be a future project.
Additionally this scheme is particularly well suited for applications in domain decomposition
methods. Especially handling with nonmatching grids often requires a treatment of boundary
conditions in a weak way. In the next chapter we will apply the scheme to the three-field method.
Part II
The three-field formulation
Chapter 3
The three-field formulation
Nowadays in the field of partial differential equations the application of nonconforming domain
decomposition methods becomes more and more important. These methods allow to use different
meshes or even discretization techniques in the subdomains. Hence it is possible to use software
tools for the discretization of the subdomain problems without taking into account the discretization of neighboring subdomains.
The most popular approach is the mortar method (cf. [Bel99], [BMP94], [Ach97]). There the
local problems are coupled by controlling the jumps across the interface. Here in part II of
the thesis we concentrate on a different method, the three-field formulation. This scheme was
introduced and analyzed by F. B REZZI and D. M ARINI in the beginning of the nineties (cf.
defined on the union of
[BM94, BM92, BBM92]). In this formulation an additional space
the local interfaces is introduced. Now the jumps between functions of and local functions defined on the subdomains are controlled.
Considering a discrete approximation of the three-field scheme certain inf-sup conditions have to
be satisfied. This is quite restrictive for practical applications. For example if we have different
meshes in different subdomains given by an automatic mesh refinement method on each subdomain, in general the inf-sup conditions are not satisfied.
One approach to remedy this drawback is given by enriching the discrete spaces by a certain
class of bubble functions. For a description, analysis, and implementation we refer the reader
to [BM00, BM01]. We choose a different way. We circumvent these conditions by appending
additional stabilization terms using ideas of [BBM92] 1 . We apply the three-field method to the
advection-diffusion-reaction equations, i.e.
" in
on
As before for the regularity of the data, we
assume (Ass. 1), i.e. ,
, and , and (Ass. 2a), i.e.
As usual
1
,
%#
,
(3.1)
with
#
, is a bounded, polyhedral domain with Lipschitz boundary.
The analysis of [BBM92] contains some mistakes [Bre02]. Therefore we use for our proofs a different technique.
42
The three-field formulation
Γ1
Γ1
Ω1
Ω3
Ω2
Ω4
Ω
Figure 3.1: Example of a domain decomposition.
It is well known that in the case of small , the singularly perturbed case, strong layers can occur
(see example at the end of chapter 1). Therefore a SUPG stabilization is added in the interior of
the subdomains and, of course, in the stability and a priori analysis the dependence on has to be
investigated.
Part II is structured in three parts. In chapter 3 we introduce the three-field formulation and prove
the well-posedness of the scheme. The stabilized three-field formulation is the subject of chapter
4. Finally, in chapter 5 we show how the scheme can be extended to the Oseen equations.
3.1 The continuous three-field formulation
First we decompose the domain
H<
We denote the interfaces by
into )
nonoverlapping subdomains
0 #
H < #
Q by
Additionally we introduce special trace spaces. We define + Q + Q Q if if We emphasize, that the outer boundary
, i.e.
is not included in the interface
(cf. Figure 3.1).
with corresponding norm
" #" O Q D F if " #" O Q D F if Q
Q
(
and inner product + . The dual of + is denoted by + . A detailed description
"#" ( + of the function spaces can be found in the appendix. The three-field formulation requires three
3.1 The continuous three-field formulation
43
different function spaces:
H < H < + Q there exists on S . Thus is given by the restriction of functions
with M& to the interface . A detailed characterization of in the context of multilevel
expansions is given by S. D
and A. K
[DK98]. The spaces are provided with
norms
Q
( Q
;
;
" " 9 H< " " > " " 9 H< " " O Q D F > (3.2)
and
" " " 1" ( #
(3.3)
Q
First the spaces and are characterized:
and
AHLKE
UNOTH
/
5
Lemma 3.1
and
are Hilbert spaces with induced norms (3.2).
### Q resp. Q for # # # are Hilbert spaces with scalar product (B.4) resp.
Also Q with # # # is a Hilbert space and so is because
(B.7). Then, by Lemma A.2, + Proof:
are Hilbert spaces. Therefore with the help of Lemma A.1 we obtain that
is a Hilbert space with norm (3.2).
)
)
of Lemma A.1.
Now we consider the norm of . To this end we need two extension operators.
Lemma 3.2 There exist continuous, linear extension operators
+ Q $#
# # #
)
Proof: (i) First we recall the local extension operators of Theorem B.9
$#
Q
the extension can be simply defined by . In the case
Q
Q from to . Then for
for Q we denote
by the extension
. By construction we
each a function can be defined by have . The continuity of is a result of
" " " " - " " Q F - " " ( +
O D In the case of
44
The three-field formulation
using (B.5) and the continuity of . The linearity of
is obvious.
(ii) Next we consider the extension . For
we define the extension
The Green’s formula yields for in
on
#
by
;
H< <
; ; %
H< H< for all multi-indices
with
.
Hence
belongs
to
. For # # # we know
because
. The
of the definition of
and therefore
continuity of is a direct consequence of the definition of " " .
Then the following norm equivalence holds for all (cf. S. B
[Ber03a], Prop.
6
+
+
+
)
ERTOLUZZA
3.2 for the seminorm).
( Q
;
"R" 9 =< " R" + > - "R" #
with there holds
Proof: Suppose . For an arbitrary " R" ( + - " N "#Q ( - "N " ( Lemma 3.3
(3.4)
by Theorem B.6 and the norm equivalence (B.5). Thus we get
; " R" ( + - ; " N " ( "N " ( H< H< ;9 " " ( + > Q H< Defining the function for " " Q "N " ( % " R" ( or by the arbitrariness of
5
(
#
Q "N " " "
,5
;9 " R" ( > Q - 9 ; "R" ( + > Q #
H< H< by Lemma 3.2 implies
The following property of the norm in (3.4) can be proved:
3.1 The continuous three-field formulation
Lemma 3.4
45
equipped with the scalar product
; =< ( + is a Hilbert space with induced norm
( Q
;
9 H< " R" + > (3.5)
(
Q resp. Q .
where + is the scalar product of Proof: Simple verification of the axioms shows that is a scalar product with induced norm
(3.5). It remains to prove the completeness. Let be a Cauchy sequence in . Then, with the
because of
help of the extension of Lemma 3.2, is a Cauchy sequence in " " ( ; " " ( H< - ;H< " " ( + " " " "
for for . The completeness of implies the existence of an element . Then with and
" " - Q 5 " 1" ( % " " ( with
we have proved the assertion.
Q
+ Q +
for the dual product on With the notation
or, if possible, the inner
product of
we are able to introduce the three-field formulation of (1.1):
Definition 3.1 The following variational formulation is called three-field formulation: Find , and
,
, such that
; H< ; H <
; H< ;
H<
;
H < #
(3.6)
46
The three-field formulation
In order to rewrite (3.6) in a more compact form, we introduce the following operators
; =< ;
+
H< 0 +
0 ; =< + is given by
with , , . Additionally, ;
H< S #
Lemma 3.5 The operators , , 0 are well defined, linear, and continuous.
+
%
(
*%
*(
%
(
Proof: Straightforward computations show that the operators are well defined and linear. The
continuity for the operator % follows from
; % ; $ K" " ( ") " ( H< =< Q
Q
% H< ( ( $ 9 ;H< " " ( > 9 ;H< ") " ( > - " R" " "
$ with continuity constants $ (cf. Lemma 1.1).
where we have used the continuity of *% With the help of the trace theorem (cf. Theorem B.6) we can estimate
; " " Q ") " ( +
%
=<
=< O D F
Q ; Q
;
- 9 H< " " O Q D F > 9 =< ") " ( > " E" :" " #
The operator 0 can be estimated in a similar way. For let be any function with
*(
;
/
/
. Then we get by virtue of the trace theorem (Theorem B.6)
0 ;
;
%
=< H< " " O
- ; H< " " O Q D F " " ( /
/
Q D F " " ( +
( Q
;
" E" 9 H< " " > - " :" " " ( 3.1 The continuous three-field formulation
and therefore the assertion with
0 47
- " E" " " ( Q
5
" E" " " #
+
+
Then the three-field formulation (3.6) can be rewritten as
% (
(
Next we show that the operator
(
0+
0
+
+#
in
in in
(3.7)
satisfies the Babuška-Brezzi condition :
Lemma 3.6 There exists a constant
0 such that
0 #
"
"" "
# # # the Riesz operator (cf. Theorem A.1)
Proof: Let be given. Defining for Q + + Q (3.8)
5
? A *(
)
we construct an element of
by
L using the extension operators
tion is satisfied because of
47 ; =
?A
>
*(
" "
# # # . The constant
0
U
H< ( +
7 ; =
( Q
7 ;=
H < "
"
?A >
? A
> L
") " U
H< ( + Q 0 H< (Q +
L H< " " ( U L H< " " ( + U H< Q
;
0 9 H< " " >
for
of Lemma 3.2. Then we observe that the Babuška-Brezzi condi
0 " E"
depends only on the shape of the subdomains
.
Remark 3.1 Note that the Babuška-Brezzi condition also holds for the bilinear form
0 " 0 " "
1 " 0 #
,5
? A !
0
:
(3.9)
48
The three-field formulation
Proof: Take
0
7 ; =
?A
>
and let
!
" "
be defined as in (3.8), then (3.9) is a result of
L
7 ; =
?A
>
L
L
7 ; =
?A
>
( +
;
9
= < " 1"
"1" ( + U Q
H<
H<
where we have used the fact that
=< ! H< " " U Q
(+
Q
(+ U
H< H< " " Q
( + > 3 " 1"
+ Q and the norm equivalence (3.4).
The following observation is very important. For a given the first two equations of (3.6) are
the variational formulation of some local Dirichlet problems where the boundary conditions are
imposed in a weak sense (cf. I. BABU ŠKA [Bab73]). We can show that the local problems are
well-posed.
be given. Then there exists a unique solution Lemma 3.7 Let
two equations of (3.6) resp. (3.7) with
" " " E" % 0 J" " "K0 " #
###
The constant 0 depends on . The solution part formulation of
in i.e.
on
on
of the first
/
/
satisfies the variational
(3.10)
(3.11)
solves
S 2 ### #
#
#
. for is given by
Q is the extension operator of Lemma 3.2.
where + )
D
(3.12)
+ Q "
(3.13)
Proof: First we prove that the first two equations of (3.7) possess a unique solution . We apply Theorem A.4: The kernel of ( is given by
;H< # # # )
*(
)
where we have used a variant of Theorem B.7. The property
! " "
*% )
*(
###
)
3.1 The continuous three-field formulation
49
follows from integration by parts and the inequality of Poincaré (Theorem B.12):
; H< ;
H< ; H< ( 3 ; H< ") " ( $#
*% #)
*(
In Lemma 3.6 we have seen that the Babuška-Brezzi condition (A.3) is fulfilled. Thus Theorem
A.4 yields the existence and uniqueness of a solution with estimate (3.10).
Now we characterize
solutions the solution. To this end we construct
, ) ,
of (3.12), define according to (3.13), and set , . Then we
solves the first two equations of (3.6), hence coincides with the solution
show that gained in the first part. The estimate
###
D
-
$ K" " ( J" " ( %
# # #
###
"B
" D F " " ( " " ( "B
" D F " " ( +
-
/
-
/
+ Q . Because of
L $ S U
S S S
# # # , we observe that is the solution of the first two equations
for all and yields
)
of (3.6).
We are now in a position to show that the three-field formulation is well-posed and equivalent to
(1.2). The proof is based on F. B REZZI , L.D. M ARINI [BM92], [BBM92]:
+
###
) , equation (3.6) possesses a unique soluTheorem 3.1 For given
, tion
. Denoting by
the solution of (1.2) with
the solutions are equivalent in the sense
H< If
additionally satisfies
Q # # # # on , is given by
Q # # # #
(3.14)
(3.15)
(3.16)
50
The three-field formulation
###
belongs to
Proof: (i) We define as in (3.14), (3.15) and observe that ) we define by
for . Next
D S " + Q Q is the extension operator of Lemma 3.2. The proof of Lemma 3.7
where + Q shows that + . With the help of Lemma 3.2 we immediately see that satisfies the
last equation of (3.6):
;
; S
H<
H< #
S
The first two equations are a consequence of the definitions of and (cf. Lemma 3.7). Hence
is a solution of (3.6).
and be two
(ii) Next we show, that the solution of (3.6) is unique. Let be !
solutions of (3.6). Then
satisfy (3.6) with
.
The second equation
of (3.6) yields
function
such that
for all # # # ) . Therefore there exists a
and on
for all (see part (ii) of the proof of Lemma 3.2). Inserting in (3.6,i) and (3.6,iii) yields
; H< ; H< Therefore we obtain and
4
#
. Considering (3.6,i) again yields
# # # and thus .
and let be defined by (3.14), (3.15),
(iii) Let be a solution of (1.2) with (3.16). It is obvious that belongs to . Moreover, because of Theorem B.11, we have satisfies (3.6). Green’s formula yields one part of the asser-.
Now we have to prove that tion:
; $ ; S =< H< ;
H< ; + H< )
*(
3.2 The discrete version of the three-field formulation
51
for , because holds in
for show that fulfills the other equations of (3.6).
###
)
. Finally simple computations
3.2 The discrete version of the three-field formulation
Now we introduce a discrete version of the three-field formulation. Using an abstract framework
we will see that the discrete spaces have to satisfy some restrictions and therefore they cannot
be chosen arbitrarily. In the next chapter we will circumvent these restrictions by stabilization.
Finally, we will introduce a finite element discretization and show the connection to the mortar
method.
Let and be finite dimensional subspaces. Furthermore, we define a
finite dimensional subspace with
H< $#
H < The following two-fold saddle point problem is considered: For a given
such that
find
;
%
=< H< ; % % (3.17)
H< ; % H< .
for all Q
+
Remark 3.2 For we can define a functional by
% + Q $#
Q
This yields an injective embedding of into + , and consequently we can identify ;
:
D
with a subspace
of <
given by
V H < # # # #
Then (3.17) can be rewritten in the notation of the previous section: For a given
, such that
% (
(
+
0+
0
in
in
in
+
+
+
+
find
(3.18)
52
The three-field formulation
Now we want to establish the existence of a unique solution to (3.18) with the help of Theorem
A.6. To this end we need some additional conditions to ensure the inf-sup conditions of Theorem
A.6. First we assume that+
+
is true for
6 ###
)
5 7 5 5 ;=
" " ") "
B 5 7 - 5 ; =
? A
>
>
(Ass. 3)
. (Ass. 3) ensures the solvability of the local problems in each subdomain.
Remark 3.3 In practice it is difficult to check the condition (Ass. 3). A criterion due to M.
F ORTIN [For77] is very useful: If there exist uniformly bounded projection operators (with respect to the discrete spaces ) satisfying
8 and if the inf-sup condition holds for the continuous case, then assumption (Ass. 3) is satisfied.
This approach is used in O. S TEINBACH [Ste00] or in S. B ERTOLUZZA [Ber03a] in order to
derive sufficient conditions for (Ass. 3).
A different technique has been developed in an article of W. DAHMEN and A. K UNOTH [DK01]
in the context of wavelet discretizations.
Remark 3.4 In S. B ERTOLUZZA [Ber03a] it is proved that if the Lagrange multiplier space is defined by the restriction of the functions of to the interface , then the inf-sup condition
(Ass. 3) is satisfied.
With the help of (Ass. 3) we can prove the following global
inf-sup condition:
+
Lemma 3.8 Assuming (Ass. 3), there exists a constant
5
5 7 5 ;=
" " " " such
that
+
*(
5 7 52;=
>
?A
>
#
8
(3.19)
Proof: cf. S. B ERTOLUZZA, A. K UNOTH [BK00], Lemma 3.3
Secondly we assume the existence of a constant 5
)5 7 C
5 ;=
such that
0 #
" " " "
5 7 5 ; =
?A
>
>
4
(Ass. 4)
Now we can apply Theorem A.6:
0 0
" " " " " " %
0 45 5 " R" 5 5 " "
where is the solution of (3.6).
Theorem 3.2 Assuming (Ass. 3) and (Ass. 4) the discrete three-field formulation (3.18) possesses
, if % is coercive on *( a unique solution . Furthermore, there exists a constant
, such that
#)
5
7 ,5
7 "
5
6 5:7 5 " (3.20)
3.2 The discrete version of the three-field formulation
Ω1
1
Vh
53
Γ1
Γ12
Γ2
Λ 1h
Φh
Λ 2h
Ω2
Vh
2
Figure 3.2: Example of triangulations corresponding to discrete spaces for the three-field formulation in
the finite element context.
The most important choice of the discrete spaces is given by finite element spaces. But for these
spaces it is very difficult to establish the inf-sup conditions (Ass. 3) and (Ass. 4). The spaces
have to be chosen very carefully. We will circumvent this problem by using some additional
stabilization terms in the next chapter. But let us first introduce some finite element spaces in
order to give an example for the discretization.
We assume that for each subdomain there exists
an admissible, shape-regular triangulation . Then we can define
( where we denote the diameter of an element and the maximal diameter w.r.t is given
by 743 .
We also define finite element spaces on the local boundaries . The shape-regular decompo sition of in intervals
for the two-dimensional case or in triangles for the three-dimensional case
is denoted by Moreover let is constant (cf. section
B.6). By . Thus for
we set
M 0 $ ,
#
be a shape-regular, admissible decomposition of . Then defining
M 0 +
we obtain the finite dimensional subspaces
H< H<
and
#
The maximum mesh size for resp. is denoted by resp. . With these three vector spaces
we have introduced a conforming finite element discretization of the three-field formulation (cf.
Figure 3.2).
54
The three-field formulation
3.3 Connection to mortar elements
Finally the connection between the
three-field formulation and the mortar method is shown for
two-dimensional domains
and geometrical conform decompositions. A decomposition is
is either empty, reduced to a common
called geometrical conform, if the intersection
vertex or to a common edge. Let us decompose the interface into pieces
where
denotes the relative interior of an one-dimensional set . Then we have
$
Q Given discrete finite element spaces
#
( $
with
and
=< we define for
a discrete space
of Lagrange multipliers
: Each interface
inherits
two triangulations, one from and one from . Now each Lagrange multiplier space
is
connected
with one of the two meshes restricted to the boundary. If, without any restriction
is called nonmortar side or slave side and the side is
of generality, we choose , the side
called mortar side. In this case we define
Q
M if contains an endpoint of where is the restriction of on .
(3.21)
Remark 3.5 The choice of the nonmortar resp. mortar side is a crucial point of any mortar
discretization, especially for discontinuous coefficients. B. W OHLMUTH ([Woh01], ch. I.5.3) has
studied the influence for the diffusion equation.
Defining an index set associated to
% %
)
by
nonmortar side of
associated with
the skeleton or global interface can be uniquely decomposed into the union of the edges of the
nonmortar sides:
5
7
5 % %
)
such that where is the set of all edges given by the finite element meshes on the boundary. Next we define
the space of Lagrange multipliers for the mortar formulation by
H <
7 D F
#
Then we can formulate the mixed mortar problem : Find such that
3.3 Connection to mortar elements
; L H< ;
55
U
D F
; ; H< D F 7
7
(3.22)
#
(3.23)
But why are the Lagrange multiplier spaces chosen by (3.21)? The reason is, that the crosspoints require a special treatment. Using the above choice it can be proved, that the inf-sup condition
+
5
5 7 5C;=
H< " "E" "
7 D F 5 7 5C;=
?A
>
>
+
(3.24)
is satisfied in adapted norms for and (cf. [Bel99], Prop. 2.6, [Woh99], Lemma 2.1).
Observing that the mortar formulation (3.22), (3.23) is also given by a saddle point problem, the
well-posedness can be proved by Theorem A.4. This is the starting point of the presentation of the
mortar method for example in F. B EN B ELGACEM [Bel99], B. W OHLMUTH [Woh99], or in C.
L ACOUR, Y. M ADAY [LM97].
In the case of a strip-wise partition we can choose the discrete spaces and of the three-field
formulation in such a way, that we receive the mortar method: The space of Lagrange multipliers
of the three-field formulation is chosen by
Thus the space
is defined by
#
#
#
H < 0
#
is a linear space and can be identified with the space . The trace space #
#
#
if is mortar side of )
and the space is defined as above.
Let us assume that 2 discrete three-field formulation (3.17) gives
;
;
H< 7 D F is a solution of (3.17). The second line of the
;
;
L
H< D F ; ; H< H< ; ; H< D F
7
7
U
56
The three-field formulation
for all . This is exactly the second line (3.23) of the mortar scheme. The first line (3.22)
is a consequence of
;
; ;
H < = < 7 D F #
So we recognize that 2 is a solution of the mortar scheme (3.22), (3.23). Vice versa it can
of (3.22), (3.23) the triple is
be shown that for a given solution a solution of the three-field formulation (3.17), provided is chosen by 2
if
is the mortar side of .
Remark 3.6 The connection of these methods is only of theoretical interest. With the above choice
of the function spaces the three-field formulation (3.17) is not well-posed, because the inf-sup
condition (Ass. 4) is violated. Hence it is not possible to control the component of the trace space
.
Remark 3.7 In general, if the decomposition is not strip-wise, there are jumps in the functions of
.
and therefore we would have a nonconforming approximation Chapter 4
A stabilized three-field formulation
In this chapter we propose a new stabilized three-field formulation applied to the advectiondiffusion equation. Using finite elements with SUPG stabilization in the interior of the subdomains our approach enables us to use almost arbitrary discrete function spaces. They need not to
satisfy the usual inf-sup conditions. We prove the stability of the scheme and an a priori estimate
which is of the same convergence order as the standard SUPG method.
4.1 A discrete stabilized scheme
The topic of this section is the discretization of (3.6). We start with any finite dimensional approximation . In order to use a finite element space for , we introduce
a shape-regular,
admissible decomposition into simplices and define in
for each subdomain
the usual way
M #
H < H< Furthermore, by we denote the restriction of the mesh to the boundary . For the discrete
and for the space we need
space of Lagrange multipliers we just impose no further assumptions.
In our analysis we will apply the following approximation result of the spaces
: There exists a
linear, bounded operator, called the quasi-interpolation operator, such that
@
( %
Q ( !)(
(
(
! "
%
(4.1)
0 0 ' , % *$ , % $ % % , &
and (cf. Lemma
. which have at least one common vertex with and - is
B.3). is the union of all
,
defined again by - 0 7 3 ( 1 - . Then we can define a global operator ### for by ,
. Moreover in Lemma 4.1 the following standard
inverse inequality will be used (cf. Lemma B.1)
B
4/ ( %
B ") " ( +
#
(4.2)
58
A stabilized three-field formulation
Unfortunately, when replacing the continuous function spaces by discrete ones, the resulting discrete scheme of (3.6) is only well-posed, if the discrete spaces satisfy the conditions
0 6
0 6
5
5 7 5 ;=
5 7 ,5C5 ;=
5 75C;=
>
?A
>
0
" " " "
H< 0
" " " "
(4.3)
>
H< 5 7 ? A 5 ; =
>
(4.4)
(cf. section 3.2). One idea to ensure the inf-sup conditions (4.3) and (4.4) is proposed in [BM01,
BM00, Buf02]. For given spaces and the authors choose the spaces of Lagrange multipliers in such a way that (4.4) is satisfied. In order to ensure the other inf-sup condition (4.3) they
enrich the space by bubble functions. But up to now this procedure is limited to linear finite
elements in and .
Here in our approach we avoid the inf-sup conditions by adding some stabilization terms following the line of [BBM92]. The stabilization of the first constraint (4.3) is standard for diffusion
dominated cases and is discussed for example by H.J.C. BARBOSA and T.J.R. H UGHES [BH92]
or R. S TENBERG [Ste95]. An application to the advection dominated case for a single domain has
been discussed in chapter 2. A stabilization of the second inf-sup condition (4.4) is discussed for
example by S. B ERTOLUZZA and A. K UNOTH [BK00, Ber03a].
In the advection dominated case there is a second problem. Using a standard discretization it is
well known that there may arise spurious oscillations of the computed solution (cf. [RST96]).
and the linear form
Therefore we use the SUPG method and replace the bilinear form
in the interior by
; ; 3
$ 3 7
7
for # # # . The
stabilization
parameter
is
defined
by
"
N
"
F for A
D
C
for % (cf. section 2.2). Then in the interior of the subdomains the
)
and by error can be measured in the streamline diffusion norm
" " ( ( M" K 4" ( ;
7 3 " " ( which gives us additional control in the streamline direction. Taking all mentioned problems into
and
account we propose the following stabilized three-field formulation: Find , such that
4.1 A discrete stabilized scheme
;
59
;
% ; H< 3 ;
3
;
% 3 H< ;
H< ;
H< +
7
3
7
for all , , moment let us only assume that
7
+
+
+
3
(4.5)
. The parameters
and
3
for will be specified later. At the
. We have used the notation
$#
on the inflow part acts only
.
outflow part M and
+
only on the
3
Remark 4.1 The requested limit behavior of the stabilization parameters
corresponds to the first choice of the parameters in chapter 2.
,
for
Before analyzing the scheme (4.5), let us shortly explain, why we have added the different stabilization terms. First let us consider
+
; 3 7
%
+ Q which are added to the first resp. third line of (4.5). The terms couple the local spaces and
the space . Especially the terms in the third row make it possible to circumvent the
+ second
inf-sup condition (4.4).
the terms in the first line ensure the boundary conditions
inFurthermore,
on the inflow part
the hyperbolic limit ( ), cf. (4.9). These terms (with
)
have been introduced by C. J OHNSON and coworkers (cf. [JP86]) for hyperbolic problems. In
domain decomposition methods this hyperbolic approach has been used by M.S. E SPEDAL, X.C. TAI, and N. YAN [ETY98] for advection-diffusion equations. But their formulation needs
additional assumptions on the direction of the flow and it yields non-optimal convergence results.
The application of these terms in the context of mortar elements is discussed in Y. ACHDOU
[Ach97] and V. B EHNS [Beh01]. Coupling / and by adding the terms
;
3
7 3 in the second row enables us to omit the inf-sup condition (4.3) (cf. [BH92], [Ste95] and
chapter 2).
60
A stabilized three-field formulation
Remark 4.2 When we insert a sufficiently regular solution of the continuous three-field
formulation (3.6) into the discrete, stabilized formulation (4.5), we observe that all additional
terms vanish. Therefore the stabilized formulation (4.5) is consistent.
Remark 4.3
Let us point out, that it is necessary to compute test functions of and on the
mesh . But such problems cannot be avoided, if finite element functions based on different
meshes are coupled. The practical implementation is discussed on page 31 in section 2.6.
Our proposed scheme also makes sense in the hyperbolic limit ( to
). Then the scheme reduces
; % ; (4.6)
=< H< ; ; (4.7)
=< 3 ; % (4.8)
H< , , . We note, that equation (4.8) determines for a given
for all . Roughly speaking is
, . Hence we can define a linear operator # # # . Inserting the operator into (4.6) yields the
given by on the outflow for scheme
; % ; (4.9)
H< H< 3
7
)
which can be interpreted as a variant of the discontinuous Galerkin scheme of C. J OHNSON (cf.
[JP86]). Then (4.7) determines the Lagrange multiplier part (which of course only makes sense if
the solution of the limit problem is sufficiently regular).
the first two lines of
Finally there is a second important observation. For a given (4.5) represent ) independently solvable, Dirichlet problems given in the
local
subdomains
.
Summarized the local problems can be written as follows: Search for such
that
(4.10)
with
; %
3 ;
3 5
; %
3 +
3
and
7
7
+
7
4.2 Analysis of the stabilized scheme
61
###
for ) . This discrete scheme is a slight modification of the proposal in chapter 2.
There the inhomogeneous boundary conditions are imposed weakly on the whole boundary
strongly. But we can derive.
Here we enforce the homogeneous boundary conditions on
a stability and an a priori estimate
completely
in the same way. Introducing the
of the scheme
following weighted norms for 3
and +
@ ( (
;
;
" " 3 E" " "R" 3 E"R" " " ( % ") " ( # # # (cf. Theorem 2.3):
we can show for 7
)
3
satisfy 3
%
3
then there holds
0 0 3
3
Lemma 4.1 If the parameters
with
7
3
(4.11)
" " @( (4.12)
where the constant
is defined by the inverse
inequality (4.2), the coercivity constant and the shape of the elements of the triangulation . Therefore the discrete problems (4.10) have
unique solutions.
4.2 Analysis of the stabilized scheme
Now the multi-domain problem (4.5) is analyzed. We start with a reformulation of the stabilized
three-field formulation (4.5). Adding the three equations in (4.5) we obtain:
and , such that
search for 2
with
; H< (4.13)
; H< ; % ; % #
3 3 +
D
8
+
4
7
7
Using this compact formulation we start our analysis by considering the stability of scheme (4.13):
62
A stabilized three-field formulation
3
Theorem 4.1 Let the parameters
satisfy (4.11). Then
is coercive:
" " 2 The norm is given by
" " ; H< " " ( " " ( ; ; % #
H< 3 #
3
(4.14)
+
7
The stabilized discrete three-field formulation (4.5) possesses a unique solution.
Proof: Suppose 2 . Then, with the help of (4.12) we compute
; H< " " ( ; ; % #
H< 3 , we obtain
Using the definition of the norm and ; H< " " ( " " ( ; ; % # (4.15)
=< 3 +
7
+
3
+
7
Because we integrate over each part of the interface twice with opposite directions of the outward
normal, the term
; ; H< 3 7
%
; %
H< vanishes. Hence, we have proved the assertion.
### Remark 4.4 The norm (4.14) has the disadvantage,
we do not control the variable .
+ that
+
+
+
) .
The norm just measures the jumps across the local interfaces for for a suitable constant ,
But if the parameters
are small enough, i.e.
we obtain
; H< 3
%
Q
" " ( " " @ ( M"" ( - " " #
(4.16)
4.2 Analysis of the stabilized scheme
63
Hence, in this case we can prove the stability of the scheme also in a norm given by the left hand
side of (4.16).
Proof: For the proof we use the trace inequality
(4.17)
% - ( ")2" ( (cf. Theorem B.8) which is valid for all , assuming that is a domain
,
with Lipschitz boundary. Then the Young inequality yields for and any
# # # E" " % %
! " " E" " #
(4.18)
Q
Now using (4.17) with and gives
Q "
"
" " D F
and . Inserting the last inequality into (4.18), we obtain
with Q #
3
F
E" " " " D E" " Choosing sufficiently near to gives the assertion.
+
)
+
+
+
+
+
+
+
+
+
+
+
+
+
Next we show a continuity estimate for the bilinear form
a priori error analysis.
. This result will be applied in the
and # # # , there holds
for " " ; H< ; 3 L U " " 0 (
;
L
U
0 3 " " " R" %
#
(
;
0 3 " " " " E" " and . Then we obtain from the definition of
Proof: Let be
:
;
L
=< ;
%
3 U #
;
(4.19)
%
0 , ,
+
- 0
Lemma
4.2 For all
,
with
)
7
7
+
7
3
7
3 3
+
3
3
+
3
7
+
64
A stabilized three-field formulation
by the Cauchy-Schwarz and the
0 (we obtain
(
" " , " " for each # # # % ; '0 ") " " " 0 E" " " R" 0
0
3 % 0 ") " ( 0 " " @ ( ; " " " R" # (4.20)
3 0
'0
Let us consider the first row of (4.19). For all
Young inequality and the definition of the norms
+ 8
)
+
3 7
+
;
7
3 D
3
7
Next, for the terms in the third row of (4.19) we get
3
3
@ ( #
@ ( ( %
0 " " 0 " " 0 " "
(4.21)
Now collecting the estimates (4.20) and (4.21), using
; 3 +
% % " R " ( 0 ") " ( 0
and adding and subtracting ! % we obtain
;
;
- 0 " " H< 0 3 L U " " 0 " " ( ;
L
U
0 3 " R" %
L ;
#
U % 3 7
3
+
+
7
7
7
3
(4.22)
The terms in the last row can be reformulated:
; ; L U %
H< 3 ; H< 7
Now, since we integrate over each part of the interface twice, we have
; H< %
#
%#
4.2 Analysis of the stabilized scheme
65
Therefore using the triangle inequality and the Young inequality, we arrive at
; ; L H< 3 - ;H< 0 ; %
3 ; ; " " ( E" " " " H< 0 3 7
7
7
U%
( "R" #
(4.23)
Applying the estimates (4.22), (4.23) yields the assertion.
In order to prove the a priori estimate we first need a continuity
estimate of the bilinear form
we get for the
. Taking into account, that the functions do
not
vanish on
) ,
streamline diffusion part the following continuity estimate (cf. (2.26)) for and arbitrary
:
# # #
0 %
( !#"
(
;
L
U
0
"
"
0
"
N
"
0 35
#
%
7
%
%
(4.24)
Now we can prove the main result of this section by combining the stability result of Theorem 4.1
and the continuity estimate of Lemma
4.2:
3
Theorem 4.2 Let the parameters
satisfy (4.11). Furthermore, let us assume that the -part of
of the three-field formulation (3.6) is sufficiently regular, i.e.
the solution H < 4
with . Then denoting the discrete solution of (4.5) by the error is bounded by
" " ;
;
- 5 5 H< L " " ( U " " 3
; ; L " " ( U ( ! (
=< 3 ( !#"
;
;
(
L " N" U 5 5 L U " " #
43 3 +
,5
7 3
7
+
3
3
7
7
+
6
,5
7 7
3
66
A stabilized three-field formulation
=< Proof: First note that we have because of the representation
and be arbitrary. Defining
(3.16)). Let , + (cf.
the consistency of the discrete formulation yields the Galerkin orthogonality
and therefore by Theorem 4.1 and Lemma 4.2 and defining " " % - 0 " " ; H< ; ; U " " H< L
0
3 ;
(
L
U
" " "
" 3 ;
" " (
"
"
3 7
%
3
7
3
+
+
7
+
3
3
E" " Now, inserting the estimate (4.24) and applying the interpolation estimates (4.1), we get
#
" " % 0 " " ; H< ; L " " ( U " " 3 0
; L " " ( U ( !)( (4.25)
3 ( !"
;
;
(
L
L
U
U
M" R"
" " 43 3
with a constant , which is independent on and . Using the definition of " " and the
interpolation estimates (4.1) we get for " "
" " ;H< " " ( " " @ ( ;
3 %
+
3
3
+
3 7
7
+
7
7
3
3
+
7
4.2 Analysis of the stabilized scheme
67
" " - ; H< ; " R" ( ( ! " " " @ ( 43 ; " " ( " " " " 3 - ; H< ; " R" ( ( !#" " " @ ( (4.26)
43 ( !#( #
;
(
L
" " " G" U
3
Then we choose 0 with the constant defined in (4.25). The triangle inequality yields
(4.27)
" " " " % " " " " #
Then the proof can be completed by inserting the estimates (4.25) and (4.26) and by taking into
account that and have been chosen arbitrarily.
In chapter 2 for one subdomain we have proposed
to choose the parameters according to
(
& ( (4.28)
7
+
7
7
+
7
+
+
+
3
+
3
3
3
3
with suitable global constants . Note that
fulfills the stability assumption (4.11). Applying this proposal the a priori estimate simplifies to
" "
- 5 5 ; H< ; L " " ( U " " 3 ; ;
(
H< L " " U ( ! (
3
( !#"
#
;
;
(
L
U
" N"
5 5 " " 3 3
5
7 7
7
7
6
5
7 7
(4.29)
If the approximation order of the discrete spaces and is sufficiently large, the error estimate
of the -part is of the same convergence order as the a priori estimate of the standard SUPG scheme
(cf. [RST96]).
Remark 4.5 In (4.28) we have chosen the hyperbolic choice. Of course it is also possible to take
the elliptic choice+ (2.37):
+
3
3
(
(4.30)
!
#
Then we get the same convergence order as with the choice (4.28). The numerical results show
that both strategies work well.
68
A stabilized three-field formulation
Mesh for Φ
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
Figure 4.1: On the left hand side the mesh in the interior is plotted ( right hand side the mesh for the global interface functions is plotted. ( 0.4
0.5
0.6
resp. )
0.7
0.8
0.9
1
). On the
4.3 Numerical results
In this paragraph we examine the theoretical results from the last section. We will see that we
really obtain the predicted convergence rates. Moreover, we give some recommendations, how to
choose the different meshes. Furthermore, we give some remarks about the implementation.
Some comments about the implementation
Again all numerical results are restricted to the two-dimensional unit square. The unit square
rectangles for arbitrary and . The different meshes for the
can be decomposed into interior of the subdomains, the Lagrange multiplier spaces and the global interface space can be
chosen arbitrarily. See Figure 4.1 for an example. In our numerical experiments the inhomogeneous boundary conditions on
are simply worked in by setting the degrees of freedom (d.o.f.)
belonging to on
to the prescribed values.
The algorithms are implemented in MATLAB
using routines of FEMLAB . In order to
accelerate the code some parts are written in the language . This has to be done especially for
parts with nested loops. The result is a speed-up of 10 to 100 depending on the mesh sizes. Main
parts of the algorithm can be parallelized. But here all computations have been done on a single
processor.
The program disintegrates into several parts (cf. Figure 4.2). We will discuss the algorithm step by
step. In the first step we decompose the unit square into rectangles and store the position
of the edges and vertices.
Then for each subdomain (rectangle)
local triangulations for and are constructed. Theoretically this can be done completely
in parallel and needs no communication between the sub
domains. The meshes for in the interior are constructed by a Delaunay algorithm (cf. [GB98],
chapter 2).
The construction of the global interface space / is more complicate. To this end first we define
an array of the global degrees of freedom (d.o.f.). Then for each subdomain an array is constructed
which maps the local degrees of freedom to the global degrees of freedom. This step requires the
communication between the subdomains.
0
4.3 Numerical results
69
1. Decomposing the global domain into subdomains
and 2. Defining local meshes for
3. Defining the global interface mesh for /
4. Building the connection
between the meshes / and
and between and .
5. Assembling all matrices
6. Solving the arising linear system
Figure 4.2: Main steps of the implementation: The steps , and can be completely done in parallel.
The next step is the assembling of the discrete system. It can be seen that the whole system can be
built up using only local information. For the matrices corresponding to the global interface space
the local array of d.o.f.’s can be used. Hence, this step can be parallelized, too.
The arising linear system is not solved directly. Instead we derive an equation for the global
interface variable, called Schur complement equation. This approach will be discussed in detail
in the next chapter. Building up the Schur complement equation explicitly would require the
inversion of local matrices. We circumvent this problem by using an iterative algorithm. Here
the GMRES method without restart is
chosen. The GMRES algorithm is stopped if the initial
. Note that each iteration step requires the solution of a
residuum is reduced by the factor local problem in each subdomain.
Results
Now the theoretical results will be illustrated by some numerical experiments. Three different
examples are presented. First we show the robustness of the algorithm even for difficult problems.
With the help of the second example the convergence properties of the algorithm are studied.
Finally we demonstrate that both parameter choices yield nearly the same results.
A first example
The main focus of our algorithm is the application to the advection dominated case. Especially
the case of nontrivial flows is of interest.
To demonstrate the power of our approach we start with
the following example in
of
Example 4.1 We search for a solution
#
#
in
on on on
70
A stabilized three-field formulation
0.5
1
0.8
0
0.6
0.4
−0.5
1
0.9
0.8
0.7
0.2
0.6
0.5
0.4
0.3
0.2
0.1
0
0
(a) stabilized three-field algorithm
(b) computation with FEMLAB
Figure 4.3: The solution of Example 4.1 for computed
in the multi-domain case using the stabilized three-field formulation (a) and with the help of FEMLAB
(b).
and 8 and the flow is
#
V V V is a rotational flow with a center in and . Furthermore, holds on the
whole boundary .
We decompose the unit square into squares. For the discretization the meshes of Figure
with defined by
4.1 are chosen. The Lagrange multiplier meshes are chosen three times finer than the interior
meshes. In the context of domain decomposition this example is particularly interesting. In the
interior the solution is almost constant in the advection dominated case. The constant is given by
the mean value of the Dirichlet data on the boundary. Now each discretization has to find this
value by mixing the boundary
information.
is plotted in Figure 4.3 (a). In Figure 4.3 (b) the solution computed with
The result for the standard algorithm of FEMLAB
is plotted (mesh size ). It can be seen that both
plots are quite similar. So it can be stated that the proposed algorithm finds the correct interface
values.
#
Verifying the -convergence
Analogously to the single-domain case we have to check the theoretically predicted convergence
rates. Again we consider the following smooth example:
Example 4.2 Let the right hand side and the boundary condition be chosen in such a way that
becomes the exact solution of
(4.31)
?
in
on
"#
4.3 Numerical results
71
error in the interior
−1
error in the interior
0
10
10
−1
10
−2
10
−2
energy
2
L (Ω)
10
−3
10
−3
10
−4
10
−4
−5
10
10
ε =1
ε =0.1
ε =0.0001
−3
10
error
−2
10
(a)
−1
−5
10
0
10
hint
ε =1
ε =0.1
ε =0.0001
10
−3
−2
10
10
−1
0
10
h
10
int
(b) energy error
error of the Lagrange multiplier
error of the interface function
−2
10
0.2
10
−3
10
0
L (Γ)
−0.2
10
2
L2(Γ)
10
−4
10
−0.4
10
−5
10
ε =1
ε =0.1
ε =0.0001
−0.6
10
−3
10
−2
10
−1
−6
0
10
hint
10
(c) Error of the Lagrange multiplier
ε =1
ε =0.1
ε =0.0001
10
−3
−2
10
10
−1
10
hint
0
10
(d) Error of the global interface function
Figure 4.4: The error of the stabilized three-field
for
formulation
mesh size. The unit square is decomposed into subdomains.
in dependence on the
In the first test case we decompose the unit square into
sub-squares. The meshes in the
or . The
interior are chosen in the same way as in Figure 4.1. The mesh size is either Lagrange multiplier meshes are always chosen three times finer than
the interior
mesh. The global
3
. For all computations+ connected
interface mesh has the mesh size with this example we
have chosen the hyperbolic choice (4.28) of the parameters with
and . Now we alter
the mesh size for . The
results are plotted in Figure 4.4. And indeed we get the
predicted convergence rates: For the -error in the interior we expected a convergence rate of in
the diffusion dominated case and of in the advection dominated case. For the interior solution
in the energy norm is expected. The error of the Lagrange
a convergence order of resp. multiplier spaced is bounded by '
in the diffusion dominated case and the error of the global
interface space is bounded by '
in the advection dominated case and ' else. Here the
#
#
#
Q 72
A stabilized three-field formulation
error in the interior
−1
error in the interior
0
10
10
−1
10
−2
2
L (Ω)
energy
10
−2
10
−3
10
−3
10
ε =1
ε =0.1
ε =0.0001
−4
10
−3
10
error
−2
10
(a)
hint
−1
−4
10
0
10
ε =1
ε =0.1
ε =0.0001
10
−3
−2
10
−1
10
0
10
h
10
int
(b) energy error
error of the Lagrange multiplier
error of the interface function
−2
10
0.2
10
0
10
−3
L (Γ)
−0.2
10
2
L2(Γ)
10
−4
10
−0.4
10
ε =1
ε =0.1
ε =0.0001
−0.6
10
−3
10
−2
10
−1
hint
10
ε =1
ε =0.1
ε =0.0001
−5
10
0
10
−3
−2
10
(c) Error of the Lagrange multiplier
−1
10
hint
10
0
10
(d) Error of the global interface function
Figure 4.5: The error of the stabilized three-field formulation in the case of subdomains for : The global interface mesh is fixed ( ) and the mesh sizes of the interior are
varied.
error of the Lagrange multiplier space is measured in the norm
Q
;
(
" " 9 H< " " A D F >
# # # H < D
and the computed global interface norm is given by
Q
;
" " ( 9 H< " 1" A D F >
$#
In the next step we test the robustness of the stabilized scheme. In the single domain case we have
seen that the algorithm is very robust with respect to the choice of the local Lagrange multiplier
4.3 Numerical results
73
error in the interior
−1
−1
−2
10
energy
2
L (Ω)
10
−3
10
−4
−2
10
−3
10
10
ε =1
ε =0.1
ε =0.0001
−5
10
error in the interior
0
10
10
−3
error
−2
10
10
(a)
hφ
−1
−4
10
0
10
10
−3
10
−2
−1
10
0
10
hφ
10
(b) energy error
error of the Lagrange multiplier
1
ε =1
ε =0.1
ε =0.0001
error of the interface function
−1
10
10
−2
L (Γ)
0
10
2
2
L (Γ)
10
−3
10
−4
10
ε =1
ε =0.1
ε =0.0001
−1
10
−3
−2
10
10
−1
hφ
10
ε =1
ε =0.1
ε =0.0001
−5
10
0
10
−3
10
(c) Error of the Lagrange multiplier
−2
−1
10
0
10
hφ
10
(d) Error of the global interface function
Figure 4.6: The error of the stabilized three-field formulation in the case of subdomains for : The interior meshes are fixed ( resp. and ) and the
mesh size is varied.
#
spaces and the local interior spaces. Therefore we uniformly refine the interior meshes and the La grange multiplier meshes keeping the global interface mesh constant ( ). The Lagrange
multiplier spaces are always chosen three times finer than the corresponding interior meshes with
resp. . Again we obtain good results. The Figure 4.5 shows this very clearly.
mesh size It can be observed that it does not make sense to use much finer meshes in the interior than for
. Only the convergence of the global interface functions in the diffusion dominated case is not
so good. This behavior is also covered by the theory, because in the second line of (4.29) there is
(
a factor , which increases for finer interior meshes.
#
# #
Now vice versa the interior meshes are fixed ( resp. , ,
) and we vary the mesh size of the global interface space. The results are plotted in
Figure 4.6.
# 74
A stabilized three-field formulation
#
#
# Next the error
in dependence
on the choice of the stabilization parameters is analyzed. To this end
+
3 , and . First the
resp.
we fix the meshes by +
3
constant
is varied for
. The results can be seen on the left hand side of Figure 4.7. On
and different is plotted. In both
the right hand side the error for fixed
cases we observe
+
that3 the scheme is quite robust with respect to the choice of the constants. Only the error of the
interface error can be significantly improved by choosing a larger value for and a smaller value
for . But on the other hand in the next chapter we will see that the required iteration steps for
the GMRES algorithm increase strongly in this case. Finally we have increased the number of
resp. , and ).
subdomains for fixed mesh sizes ( Therefore we have decomposed the unit square into sub-squares and have computed the
error for different . In Figure 4.8 we observe a slight increase of the Lagrange multiplier error.
But this error is simply caused by the fact, that the interface also grows.
#
#
# Hyperbolic choice versus elliptic choice
Now we compare the two different proposed choices in the advection dominated case. For the hyperbolic choice is given by
+
+
+
and the elliptic choice by
+
3
3
3
+
3
3
(4.32)
(4.33)
with constants . Both parameter strategies are tested in the following example due to
E. B URMAN and P. H ANSBO [BH02]:
Example 4.3 Let the right hand side and the boundary condition be chosen in such a way that
# # (4.34)
becomes the exact solution of
with .
in
on
#
The solution is constant in the direction and possesses a sharp layer at
. Now we
decompose the domain into -rectangles. This case is interesting because the terms with
vanish on the part of the interface where the layer of the reference solution is located. Again
the mesh sizes in the interior are chosen by a checkerboard pattern. In Figure 4.9 (a) the solution
(4.34) is plotted with the elliptic choice of the stabilization parameters. On the right hand side we
can see that both parameter strategies give nearly the same (optimal) results.
4.3 Numerical results
75
error in the interior
−3
10
L (Ω)
−4
10
2
2
L (Ω)
error in the interior
−3
10
ε =1
ε =0.1
ε =0.0001
−5
10 −3
10
−2
−1
10
0
10
α
1
10
10
−4
10
error
ε =1
ε =0.1
ε =0.0001
−5
10 −3
10
2
10
−2
−1
10
10
0
10
β
1
10
2
10
error of the Lagrange multiplier
error of the Lagrange multiplier
ε =1
ε =0.1
ε =0.0001
−0.47
10
−0.48
2
L (Γ)
10
0
10
−0.49
10
ε =1
ε =0.1
ε =0.0001
L2(Γ) −0.5
10
−3
−1
−2
10
10
α
10
0
1
10
10
−3
2
10
10
−2
−1
10
10
0
10
β
1
10
2
10
Error of the Lagrange multiplier
error of the interface function
−3
−4
2
10
−5
10 −3
10
error of the interface function
−3
10
ε =1
ε =0.1
ε =0.0001
L (Γ)
2
L (Γ)
10
−4
10
ε =1
ε =0.1
ε =0.0001
−5
−2
10
−1
0
10
10
1
10
2
10
10 −3
10
−2
−1
10
10
alpha
0
10
β
1
10
2
10
Error of the global interface function
Figure 4.7: The error of the stabilized three-field formulation in the case of subdomains for : The meshes are fixed ( resp. , , ). On the
left hand side the parameter is varied and on the right hand side the parameter .
76
A stabilized three-field formulation
energy error in the interior
−1
error of the Lagrange multiplier
1
10
10
−2
10
0
L (Γ)
−3
10
2
energy
10
−1
10
−4
10
ε =1
ε =0.1
ε =0.0001
−5
10
0
2
4
6
8
Number of subdomains
10
ε =1
ε =0.1
ε =0.0001
−2
10
12
0
2
4
6
8
Number of subdomains
10
12
(b) Lagrange multiplier error
(a) energy error
Figure 4.8: The error of the stabilized three-field formulation for and fixed mesh
sizes
resp. , and ): The domain is decomposed
( into
sub-squares for different . Now the error is plotted for different .
error in the interior
−1
10
1.2
1
−2
10
energy
0.8
0.6
0.4
−3
10
0.2
0
hyberbolic choice
elliptic choice
1
−0.2
0
0.8
0.2
0.6
0.4
0.4
0.6
0.8
0.2
1
0
−4
10
−3
10
−2
10
−1
hint
10
0
10
(a) Solution of Example 4.3
(b) Comparison of the stabilization strategies
resp. , ,
Figure 4.9: In (a) the solution of Example 4.3 is plotted for , where , are chosen according to (4.33). In (b) for different mesh sizes ( resp.
, resp. and ) we observe the energy error in the interior for
the two different parameter choices.
4.4 Conclusions
We have proposed a scheme, which is stable independently of . Furthermore, the discrete function
spaces could be chosen almost arbitrarily. It was only requested, that in the inside of the local
subdomains a shape regular, finite element discretization is imposed. For this scheme, we could
determine the free parameters in such a way, that the -part of our scheme possesses the same
error order as the standard SUPG-method. We are not aware of any other multi-domain stabilized
scheme with optimal convergence rates in the singularly perturbed case. This scheme possesses a
4.4 Conclusions
77
wide range of applications. Especially the extension to the Navier-Stokes equation is an interesting
task for further research.
The numerical experiments confirm the predicted convergence rates. Moreover, it could be shown
that the algorithm is very robust. It also works quite well for examples with strong layers. Finally
the number of subdomains does not deteriorate the convergence speed.
We were surprised, that the algorithm always works better in the singularly perturbed case than
in the diffusion dominated case. One reason is, that due to the advection term the information is
transported.
Chapter 5
The three-field formulation for the
Oseen Equations
The subject of this chapter is the extension of the theory of the three-field formulation for scalar
elliptic equations to the Oseen equations. Here, due to the additional constraint that a vector field
must be divergence free, additional problems occur.
In the first section we describe the Oseen equations and derive their weak formulation. Then we
formulate and analyze a three-field formulation for these equations. We will restrict our discussion
solely to the continuous case.
5.1 The Oseen equations
The Oseen equations can be considered as a linearized version of the Navier-Stokes equations.
In many implicit schemes for the time dependent Navier-Stokes equations a sequence of Oseen
equations has to be solved.
The Oseen equations are given by
.
H <
in
in
on
(5.1)
where we have imposed homogenous boundary conditions for simplicity. As usual
,
, is a bounded domain with Lipschitz boundary. Here we search for a velocity field
and pressure . is a given prescribed velocity field, is a reaction
coefficient and is the viscosity. Within an implicit scheme of the time dependent Navier-Stokes
equations is given by the inverse of the time step. For the data we assume
. H <
.
"
. with .
#
In order to derive the variational formulation of (5.1) we define bilinear forms
.J ; =< .
(Ass. 4)
80
The three-field formulation for the Oseen Equations
, . is equipped with the usual norm
;
(
#
;
" " H< " " (
Further norms " " and are given analogously.
Defining the function spaces for the velocity and ! for the pressure the weak form of
(5.1) is given by
Find
. H < (5.2)
#
. for a subdomain
and
$
*
$
$
Problem (5.2) is again a saddle point problem. And therefore we can apply Theorem A.4. This
yields the following result:
Theorem 5.1 Under the assumption (Ass. 4) there exists a unique solution
(5.2) and the solution depends continuously on the data in the sense
*
$
of
0 6 " 1" ( " '"B % 0 " 1" #
Proof: (i) It is a well known result, that the bilinear form . satisfies the inf-sup condition
. #
6
0
0
(5.3)
" " " "
5
7:9 ; =
/
; =
> B27 ? A >
(cf. V. G IRAULT, P.A. R AVIART [GR86], ch. 1, 5.1).
(ii) The ellipticity is a consequence of integration by parts and the inequality of Poincaré
(cf. Theorem B.12)
( " '" ( 3 " 1" ( #
Applying Theorem A.4 yields the assertion.
5.2 The three-field formulation
The extension of the three-field formulation from the advection–diffusion problem to the Oseen
problem is not straightforward, because the correct treatment of the pressure and the divergence
constraint cause some new problems.
Let us start with a decomposition of the domain into ) nonoverlapping subdomains . Each
subdomain
should possess a Lipschitz boundary. We can define the following function spaces
analogously to the advection-diffusion problem :
H < H < + Q (5.4)
5.2 The three-field formulation
81
#
and
The corresponding norms are natural extensions of the scalar case and are given by
(
;
" " H< " " ;
" " < " " ;
" N" < " " (5.5)
(5.6)
(5.7)
" "
" #"
where and are defined in the previous chapter for the scalar three-field formulation.
All function
and
spaces are Banach spaces with the above norms. For elements the dual product is defined by
#
;
;
H< < 0
The correct choice of the function space for the pressure is more difficult. The space
H < is too large to ensure a unique solution. Therefore, imposing that the mean values of the elements
are zero, we consider the following subspace for the pressure:
;
#
#
#
H< #
Remark 5.1 The drawback of this choice is the global coupling of the function space . The
idea of the three-field formulation is a preferable local choice. Only then it is possible to assemble
the matrices separately in the discrete case. In addition, the derivation of domain decomposition
methods is difficult. Therefore later on we will give some approaches to circumvent this coupling.
Now the three-field formulation of the Oseen equations can be formulated by
82
The three-field formulation for the Oseen Equations
. . Find ; =<
; =<
<
; ; H< < ### (5.8)
) is used for where the notation
for . Due to the additional
pressure space we should rather speak about a four-field formulation. But in order to demonstrate
the close connection to the scalar case, we also call it a three-field formulation.
The next step is the proof of the main result:
and
Theorem 5.2 The three-field formulation (5.8) possesses a unique solution .
$ the solution of the global problem (5.2), the solutions Denoting by $ are equivalent in the sense
(5.9)
(5.10)
#
(5.11)
and , is given by
If additionally satisfies #
(5.12)
Proof: Let be a solution of (5.2). Now we define
, and by (5.9), (5.10) and
is more
(5.11). The construction of the Lagrange multiplier involved. For an element
Q
+ we define
.
& + Q is the extension operator of Lemma 3.2. It is easily seeing that
where
+ Q and therefore we have .
Moreover it is obvious that (5.8,
ii),by(5.8, iii) are satisfied. In order to show (5.8, iv) we define the
global operator in #
on $
5.2 The three-field formulation
83
This operator is well defined (cf. Lemma 3.2). Then we compute
; ; < H< ; < . for # # # . In the last step we have used that < Let us also define is a solution of (5.2) and by
in
###
# # #
. Then the definition of )
#
yields
;
. < H< ; . ; H< < . ; < ;
for
Because of the construction of the extension operators
, for each
. Hence, (5.8, iv) is satisfied.
It remains to prove (5.8, i).
we define an element
& ;
<
; . $ H< . ; < ; $ . ; H< < for arbitrary . In the last step we have used the fact, that satisfies (5.2). This ends the
existence part of the theorem.
Now we consider the uniqueness. Because (5.8) is a linear system it is sufficient to investigate the
homogenous system with and to show that the trivial solution is the only solution.
Let us start again with (5.8, iii). Due to this equation there exists an element
such
that
#
(5.13)
84
The three-field formulation for the Oseen Equations
Using
as test-function in (5.8,i) and using (5.8,ii), (5.8,iv) yields
;
.
H< ;
$#
H< $ Taking into account the ellipticity of we arrive at , hence , (cf. (5.13)).
In order to determine the pressure we define a global pressure such that .
Then (5.8,i) yields
. ; H< . #
4
By virtue of the inf-sup condition (5.3) we obtain that we obtain for all . This implies .
, thence
. Using (5.8,i) again
+ Q integration by parts yields
. . #
It remains to prove the representation (5.12). For
Using that
Theorem 3.1).
is the solution of (5.2) we obtain the assertion by a standard argument (cf.
As already mentioned the critical point is the choice of the pressure space
the space by and formulate the additional condition:
. One idea is to replace
; =< #
With the help of this additional constraint it can be proved again that the three-field formulation is
well-posed.
Finally let us briefly discuss the discretization. Principally we can use the same stabilization terms
as in the scalar case in order to circumvent both inf-sup conditions. The technique to circumvent
the additional inf-sup condition for the pressure-velocity coupling is well known (cf. [Lub94],
[FF92]). Therefore it seems to be possible to extend our stabilized scheme to the Oseen equations.
However, a detailed presentation is the subject for further research.
Part III
Nonoverlapping domain decomposition
methods
Chapter 6
A preconditioned Schur complement
method
In the last part we introduced and analyzed a multi-domain formulation for the advection-diffusion
equation. In part III of the thesis we explain, how the system can be efficiently solved in parallel.
To this end the global problem is decoupled into a sequence of local problems. We present two
different algorithms: A Schur complement method and an alternating Schwarz algorithm. In this
chapter we present the former method. The alternating Schwarz method is presented in chapter 7.
Finally, in chapter 8 we compare both algorithms.
Let us start with the Schur complement method. The key idea for this method is to eliminate the
degrees in the interior of the subdomains. The remaining equation, defined on the interface, is
called the Schur complement equation. In the continuous case the corresponding operator is called
the Steklov-Poincaré operator.
The interface equation is not well conditioned. Therefore we introduce a preconditioner consisting
of a weighted sum of inverses of local Steklov-Poincaré operators. Some numerical experiments
show the effect of the preconditioning.
6.1 The continuous case
First we consider the continuous case. We derive the Steklov-Poincaré operator from the threefield formulation (3.6). Then we introduce the Robin–Robin preconditioner and discuss some
important properties. Moreover, the Richardson iteration of the preconditioned equation is considered and a differential interpretation of this iteration scheme is given.
Steklov-Poincaré operator
Let us start with the three-field formulation: Find % (
(
+
+0 0
in
in in
+
+
+
, such that
(6.1)
88
A preconditioned Schur complement method
+
using the compact formulation which was introduced in (3.7). In order to clarify the structure of
(6.1), we define an operator
by
%
(
+
(
By virtue of Lemma 3.7 is an isomorphism. The action of
local problems (3.11) with Dirichlet data on the interfaces
,
by
#
corresponds to the solution of the
. Furthermore,
we define operators
+ + +
$#
Let be the solution of (6.1). Taking
+
+ 0 0 into account, we derive from (6.1)
0 0 + 0 0 + 0 0
or
0 (6.2)
with
0 0 + $#
Since the last equation is of great importance in the theory of domain decomposition methods, it
has its own name:
+
0
0
Definition 6.1 The operator
0+ is called the Steklov-Poincaré operator and equation (6.2), given by
is often called the Schur complement equation.
Remark 6.1 Let
. Using the definition of the operator
0
;
=<
! (6.3)
0
, i.e.
and the representation (3.13), the right hand side of (6.3) can be written as
;
;
H< H< 0
; H<
S S S $ 6.1 The continuous case
89
# # # $ where
is an arbitrary function with
of the Dirichlet problems
###
and is the solution
for ) (cf. Lemma 3.7). Hence the right hand side can be computed by solving local
problems in parallel, too.
The next remark shows the close connection to the theory of linear mixed problems:
Remark 6.2 Defining the operators
+ and + +
0 , the three-field formulation (6.1) can be written as
for all + in +
+
in
,+ is given by
where #
!
*%
*(
*(
by
(6.4)
Hence, the three-field formulation can be seen as two coupled linear mixed problems. Therefore
an alternative proof of the well-posedness of the three-field formulation can be derived with the
help of the theory of linear mixed problems.
Now we prove that
is a bijective operator from
+
onto
+.
+
Theorem 6.1
is continuous and there exists a constant
+
diffusion coefficient such that
Thus
is an isomorphism from
Proof: Let
" " #
!
be given. We define
D
In addition, we know
0 6 depending on the
(6.5)
0 + #
by
# # # is given
2 ### and by
+ Q Note that +.
onto
"
# # #
) # # #
" R"" " % 0 "K0 + 1 "
/
)
#
90
A preconditioned Schur complement method
0 +
+
0
0
" 1" "K0 0 + 1" K" 0 E" % 0 "K0 " "K0 + " - 0 "1" #
. We obtain
For there exists such that 0 ; ; H< =< where we have used that satisfies the first equation of (3.6). Defining by for # # # yields by virtue of Lemma 1.1 and the definition of " &"
in
; $ " ' " ( " " #
H< (cf. Lemma 3.7).
is the extension operator of Lemma 3.2. Then, using
and the continuity of resp.
, we obtain the continuity of
)
!
Considering Lemma 3.3 we observe that the properties also hold in an equivalent norm, which
is induced by a scalar product (cf. Lemma 3.4). Thus the Lax-Milgram Lemma (Theorem A.2)
is an isomorphism.
yields that
+
Remark 6.3 The last Theorem shows that the Steklov-Poincar é operator
;
H<
;
H < is any function with 0 + . where
can be represented by
(6.6)
and is given by
Moreover, the dependence on in (6.5) reflects the behavior of the local problems, where strong
layers can occur in the advection-dominated regime. Therefore preconditioning is advisable for
any iterative solver of (6.3).
Now we introduce the local Steklov-Poincaré operators. To this end we rewrite (6.6) by
; ; % =< H< < Q Q #
#
#
we define + + by
Definition 6.2 For % + Q is given by Corollary 1.1 and is any extension with .
where
(6.7)
and define:
)
(6.8)
6.1 The continuous case
91
The local Steklov-Poincaré operators are well defined, since for we have
<
D
with $ taking into account
and the definition of . Hence the global SteklovPoincaré operator can be written as the sum of local Steklov-Poincaré operators
; #
(6.9)
H< D
D
D
D
.
The representation (6.9) clearly shows the local character
of the Steklov-Poincaré operator. Fur
thermore, it can be proved that the local operators
are continuous and coercive:
0 " " O Q D F %
Lemma 6.1 There exist constants
0 , 0 depending on such that
%
0 " " O Q D F " " O Q D F Proof: (i) First, let us point out again that
as
can be written
%
is given by Lemma
where ( 3.2 with
" " - " " O Q D F #
(ii) Taking into account the a priori estimate
" " ( % 0 " " Q F
O D
(cf. Corollary 1.1), the continuity of
by %
can be derived
- 0 " " ( J" " O Q D F " "BA D F "
% 0 " " Q F " " Q F #
O D O D
/
/
/
+ Q $#
/
(6.10)
/
/
/
"BA D F
/
(iii) The coercivity is a result of integration by parts, and the trace inequality (Theorem B.8):
#
(
3 0 " " 0 " " O Q D F
/
Thus the local Steklov–Poincaré operators
can be inverted and we obtain:
92
A preconditioned Schur complement method
: For + Q Remark 6.4 The computation of the inverses of the local Steklov-Poincar é operators corresponds
to the solution
of local problems with a Robin condition on the interface
search for
such that
Then
%
& #
(6.11)
. The equation (6.11) is the variational formulation of
is given by
in
on # on The problem (6.11) is well-posed and therefore
has a continuous inverse.
The Robin–Robin preconditioner
In the last section we have seen that the Steklov–Poincaré operator is poorly conditioned. The
continuity and the coercivity constant depend on a negative power of . In the discrete symmetric
case it can be shown that the condition also depends on
the mesh size and the maximum of
( , cf. [Bre99]). Therefore in practice
the diameters of the subdomains
'
preconditioning is mandatory.
Here we focus on a special preconditioner. The preconditioner has been developed by the group
of Y. ACHDOU , P. L E TALLEC , F. NATAF ET AL . ( cf. [AJT 99], [Nat99], [AN97], [GTN03]).
It is a generalization of the Neumann-Neumann preconditioner, which can be applied if the appropriate bilinear form is symmetric. In the discrete case the Neumann-Neumann preconditioner has been well investigated in the last years (cf. J. M ANDEL [MB93], [Man92], T.F. C HAN,
T.P. M ATHEW [CM94], B. S MITH, P. B JORSTAD, W. G ROPP [SBG96] or O. W IDLUND [DW95],
[DSW94]). For some analysis in the continuous case we refer to the work of Y.H. D E ROECK and
P. L E TALLEC [RT91].
As a preconditioner we use a weighted sum of inverses of local Steklov-Poincaré operators. Thus
the proposed preconditioner is defined by
+
;
Here
where
+ Q +
=<
#
(6.12)
are linear, bounded operators with
;
+ Q is given by H< for (6.13)
###
)
.
6.1 The continuous case
93
Remark 6.5
Usually, the linear operators
quired that
are constructed in a simple way. In many cases it is re-
and " + " " "
(6.14)
Q + , with constants .
for all + , # # # the application of requires the solution of a local problem
For each ;
)
with Robin conditions on the local interface . Therefore the preconditioner is called
on , the Robin condition becomes
Robin-Robin preconditioner. In the case of a Neumann condition. Thus the preconditioner degenerates to the well-known NeumannNeumann preconditioner in the symmetric case .
Now we show, that the preconditioner is continuous and positive definite.
+
3
+
+
$ Lemma 6.2 Let be a Hilbert space and
the dual space of . Furthermore, let
be linear, continuous with constant
, and coercive with constant . Then
exists and is
linear, continuous with constant , and coercive with constant .
$
$
Proof: The existence and continuity
of
follow from the Lax-Milgram Lemma. It remains
to prove the coercivity of
. This follows from
3
" "
$
%
" $ $
"
%
$ $
B$
/
3
+
/
+
"$
3
+
"
B$
3
In Lemma 6.1 it was proven, that the local Steklov–Poincaré operators
and , such that
coercive, i.e. there are positive
constants
3
+
% " " Q " " Q O D F O D F
+ Q /
/
+
for all . Note, that the constants
enable the following theorem:
Theorem 6.2 Let us assume, that the operators
is continuous and coercive.
and
3
and
+#
are continuous and
" " O Q D F
/
may depend on . These properties
satisfy (6.14). Then the preconditioner
9 ;H< + >
94
Proof: Let
lemma with
A preconditioned Schur complement method
+
$ be given. Then, taking into account the properties of
, we obtain
;
;
H< H< +
+
+
+
and the last
; H< " + " O Q D F
#
;
H< " "
3
+
/
3
The continuity of /
is a simple result of the continuity of the operators
This shows that the operator , ###
)
.
is well-posed.
A Richardson Iteration
Next a Richardson iteration is applied to the Schur complement equation (6.3). In the discrete case
we replace the Richardson iteration by a Krylov subspace method. The advantage of the Richardson iteration is its simple structure, which can be better analyzed. But in practical computations a
Krylov subspace method is more robust and gives better results.
The preconditioned Richardson iteration of the Schur complement equation is given by
with a damping parameter
(6.3).
Find
.
(6.15)
denotes the right hand side of the Schur complement equation
Differential interpretation
Taking into account the definition of the Steklov-Poincaré operator and the preconditioner it
is possible to derive a differential interpretation of the Richardson iteration (6.15). Defining the
and by each iteration step of (6.15) can be written in a
interface between
different way (cf. L.C. B ERSELLI, F. S ALERI [BS00]): Given
, for ) solve
the following local problems with Dirichlet data on the interface:
in
on
on
"#
###
6.1 The continuous case
Then for ###
)
95
compute the mixed problems with Robin data on the interface
+ L + U
and update the interface function by
+ Q ;
H< for ###
)
on
in
on
+ Q where
can be any weighted extension of
to by zero satisfying (6.13).
This method involves the solution of two local problems on each subdomain at each iteration step.
Hence, it can be interpreted as an iteration-by-subdomains method.
A convergence result
+ Q $ Q + . Most of them are based on the representation of the book of A. QUARTERONI, A. VALLI
In this paragraph we show some results in the case of two subdomains, i.e.
( cf. [QV99], ch. 5.1 ).
It will be shown that the Richardson iteration of the preconditioned Schur complement equation
converges linearly, if the diffusion coefficient is not too small. Without any restriction we assume
that the preconditioner can be written as
+
+
with constants
. We know from Lemma 6.2, that coercive. That means in particular, that there are constants
depending on , such that
3
for all % "1" " " !!
+
"1"
3
3
are continuous and
,
. Furthermore, we use the following convergence theorem (cf. [QV99], ch. 4.2 ):
Theorem 6.3 Let
and coercive
with
constants
and Then there exists a
3
+
+
#
! '" " converges linearly in
, such that for all
be the local Steklov-Poincaré operators, which are continuous
and . Assume the existence of a constant , such that
satisfies the condition
to the solution
(6.16)
and for any given
of the equation
.
the sequence
(6.17)
96
A preconditioned Schur complement method
3 By
"#" 3
Proof: With
the help of the continuous and coercive operator part of defines a scalar product
we observe that the symmetric
#
Q
3 3 . The properties of "
"
" " % " "3 % " " #
+
we denote the corresponding
norm
3
yield
To prove the convergence of the iterative scheme (6.17) it is sufficient to show that the map
%
%
" "
is a contraction with respect to the norm 3 , since then we can apply the fixed-point theorem
of Banach. Let
and
. We obtain
" " 3 % " " 3 " "3 '" " where we have used the property (6.16). Because of the continuity of L QQ U and we have
% " " #
%
3
3
+
" " 3 % @" " 3
%
% +
+
with constants
+
3
3
+
L Q
Q
U 3
An easy computation yields, that % is a contraction for all
3
3
3
and
3
Inserting this, we obtain
3
+
with
#
with
#
Now we investigate the meaning of the condition (6.16). Unfortunately we will see, that the condition is only fulfilled in the diffusion dominated case. can be deFirst let us point out that the bilinear form
L U
composed into a symmetric and a skew-symmetric part:
with
% 6 and
6.2 The discrete case
for . Assume
and for , we obtain Thus we get
+
Therefore denoting by
;
H< %
+
;
H< " "
" R"
#
*- #
-
( (
" " " (
H< 0 " " where we have used the continuity of
and on . Thus we obtain for
=< ( " N" ( 0 %
- - *- - - and - and - .
- - #
given in (6.5) we can conclude
H< ; " R" ( " ;
and
. Defining
the coercivity constant of
Then, using Corollary 1.1, we can estimate
97
+
. The constant
0
depends
(6.18)
that the condition (6.16) is satisfied. So we have proved the following result:
Theorem 6.4 Suppose that (6.18) holds true. Then the Richardson iteration converges linearly to
the unique solution of the Schur complement equation in the case of two subdomains.
Condition (6.18) is satisfied, if the skew-symmetric part is sufficiently small compared to the
symmetric part. Unfortunately, this is only satisfied in the diffusion dominated case.
6.2 The discrete case
In the last section the Steklov-Poincaré operator was analyzed in the continuous case. Now we
perform the same steps on the discrete level. Starting with the discrete stabilized scheme for the
three-field formulation we will derive the corresponding discrete Schur complement equation. The
solution of this equation can be obtained by an iterative decoupling of the global problem into local
problems. The computation of the local problems can be done completely in parallel. Moreover,
98
A preconditioned Schur complement method
we give some remarks, how to build a suitable preconditioner for the discrete Schur complement
equation using the ideas of the last section.
Let us first recall the stabilized formulation: Find and , such that
; H< ;
% ; H< 3 ;
;
H< 3
;
;
H< % 3 +
7
3
8
7
7
+
(6.19)
, . In this section we assume that the stabilization parameters
, the first two lines of (6.19)
are local Dirichlet problems.
, we denote the solutions (6.20)
with
; 3 %
;
3 3
, for all satisfy (4.11).
Recall that for given By ) of
+
7
3
and
7
; 3 +
%
# # # (cf. chapter 4). Then, due to the linearity of the scheme (6.20), we see
for ) # # $#
7
)
Inserting this in the third line of (6.19) yields the Schur complement equation for our stabilized
, such that
scheme: Find 2
4
8
(6.21)
where the discrete Steklov-Poincaré operator is defined by
; H < 8
6.2 The discrete case
99
and the right hand side is given by
; H < is defined by
#
% + Q $#
;
3 solves (6.20) we deduce
Taking into account that ;
H< ;
H< ;
3 +
7
3
for all 2
7
. Using (6.20) again we obtain
; H< ;
3 3
7
which can be expressed by the bilinear form of the stabilized three-field formulation (cf. (4.13))
# # $#
8
Now, using the stability of the scheme (cf. Theorem 4.1), we obtain
" # " #
(6.22)
Since the right hand side of (6.22) really defines a norm on
, we have proved:
Lemma 6.3 The discrete Steklov-Poincaré operator is elliptic with respect to the norm
" " " # " where the coercivity constant does not depend on . Therefore the discrete Schur complement
.
equation (6.21) possesses a unique solution Moreover, ) is the solution of the discrete three-field formulation (4.5).
100
A preconditioned Schur complement method
In a next step we show that the operator can be written as a sum of local operators, which are
coercive in a corresponding local norm. Starting with the definition of we obtain
; ; % H< 3 ; =< %
; =<
4
4
+
7
for 2 we obtain . Because of the opposite signs of the outward normal vectors
;
H<
;
%
H<
and
on
%
for the terms in the last line. Hence we can write the operator as the sum of local components:
with
for ; 3 +
7
.
H<
;
%
%
(6.23)
Remark 6.6 Note, that the applications of really correspond to the solution of local problems.
Thus it can be done completely in parallel. First, we have to solve a local Dirichlet problem in .
(plus the weighted approximation error of the solution
Then we give back the Robin values on
of the Dirichlet problem on the boundary).
Moreover, we can show that the operators are coercive.
. Then there holds 3
Lemma 6.4 Assume " " ( " " ( ( #
%
" " 6.3 Numerical results
101
. Then, using the definition of Proof:
Assume , we see
; 3 ;
3 3
+ 7
7
Now, applying Lemma 4.1 yields the assertion by
; 3 7
+
@ ( "
"
3
%
% #
%
" " ( " " ( % " " ( #
As already mentioned the application of a suitable preconditioner mostly improves the performance of the linear solver. Furthermore, it should be possible to compute the application of the
preconditioner in parallel. Analogously to the continuous
case we propose a preconditioner which
is built up by a sum of approximate inverses of the local operators
following the ideas of
[BS00, ATNV00]. Unfortunately
it is too costly to compute exactly. Instead, we use a
direct discretization of
as an approximation (cf. (6.11)).
Of course, in the literature many other preconditioners are discussed. For example the application
of the BPS-preconditioner ([BPS86]) in the context of the three-field approach can be found in
[Ber00a]. But a detailed description of such preconditioners and numerical results are the subject
for future work.
6.3 Numerical results
This paragraph contains some numerical results concerning the Schur complement equation and
its preconditioning.
Implementation
The main focus is an efficient solution of the discrete Schur complement equation (6.21). In the
previous chapter we explained, how all matrices are assembled. As already explained it is too
costly to build up the Schur complement matrix explicitly. Instead, we use an iterative algorithm
in order to solve the Schur complement equation (6.21)
8
4
#
(6.24)
102
A preconditioned Schur complement method
Then we only have to be able to evaluate the action of . The application requires the solution of
local problems in each subdomain. The local problems are solved by the standard direct solver of
MATLAB . We choose the GMRES algorithm as the iterative solver for the Schur complement
equation. The GMRES method is a Krylov subspace method. In contrast to many classical iterative
methods like Jacobi, SOR or SSOR, the GMRES method works quite robustly. As a motivation of
the GMRES method we mention the following property, which justifies the name of the algorithm:
Remark 6.7 The -th step of the Generalized Minimum Residual Method (GMRES) is given by
the solution of the discrete minimum problem
where
' " # # # 5
7 "
is an initial guess and
with
. is called the -th Krylov subspace. For a practical introduction into
the field of Krylov methods compare Y. S AAD [Saa96, SS86]. Here the GMRES method without
restart is used, since for our test cases the number of degrees of freedom of is not too large.
In the diffusion dominated case preconditioning of the discrete Schur complement equation (6.24)
is mandatory. Since it is too
the
costly to invert the operators
we use for the preconditioner
direct discretization
. Thus we define for the operator by of
where satisfies
%
%
#
(6.25)
Remark 6.8 Note that (6.25) is the discretization of the local Robin problem
Now we define the preconditioner by
; H <
%#
in
on
on
#
(6.26)
to the local interface . In order to implement the precon restricts the functions ditioner the solutions of (6.25) must be transferred to the grid of / . This is done
by the operators . In our implementation the operators are realized with the help of local
-projections.
is a diagonal matrix. The entries of are given by the reciprocal of the
number of subdomains to which the corresponding degrees of freedom belong.
GMRES method without preconditioner
First we consider the case without preconditioner. The experiments will show that the number of
iteration steps increases for smaller mesh sizes and for a larger number of subdomains. Only in the
advection dominated case the number of iteration steps is mostly independent of the mesh sizes.
6.3 Numerical results
103
1
0.05
19
18
54
56
hyperbolic choice
0.02 0.01 0.005
39
41
54
28
37
53
56
51
48
51
47
51
0.05
19
18
26
26
elliptic choice
0.02 0.01
39
41
28
37
42
63
43
67
0.005
54
53
94
105
Table 6.1: Number of iteration steps of the GMRES algorithm, which is needed for different mesh sizes
. The method is applied to
and diffusion coefficients
to reduce the initial residuum by the factor
Example 4.3 for a partition.
Dependence on the mesh size
We start with Example 4.3. For this test case we decompose the unit square into rectangles.
Since the flow is parallel to parts of the interface, there the information is only transported by
the diffusion part, which is small in the singularly perturbed case. We tested the hyperbolic choice
3
3
+
+
(
& (6.27)
(
and the elliptic choice
+
+
!
(
3
3
(6.28)
of the parameters. For a global mesh size the local meshes are chosen by a checkerboard pattern
with local mesh sizes
#
+
(6.29)
3
In Table 6.1 the number of iteration steps is printed, which is needed to reduce the initial residuum
.
by the factor . The initial guess is always and we always use
In the diffusion dominated case ( ) the hyperbolic and the elliptic choice are equal.
We observe that the number of iteration steps increases for finer meshes. The reason is that the
convergence of the GMRES methods depends on the condition number. The asymptotic behavior
of the condition number is given by
#
$#
(6.30)
in the diffusion dominated case (cf. [Bre99]).
is the maximum diameter of the
. A different behavior can be observed in the singularly
subdomains
perturbed case ( ). For
the elliptic choice we observe the same behavior as in the diffusion dominated case, while for the
hyperbolic choice the number of iteration steps is almost constant. This is caused by the direction
of the flow . Since the information is not transported across the whole interface, parts of the
interface are only linked together by terms corresponding to the diffusion term. For the elliptic
choice the Dirichlet values on the interface are weighted too strong in comparison to the Neumann
values.
104
A preconditioned Schur complement method
1
0.05
21
19
19
19
hyperbolic choice
0.02 0.01 0.005
32
46
66
31
43
59
20
19
18
20
19
19
0.05
21
19
17
17
elliptic choice
0.02 0.01
32
46
31
43
17
17
17
18
0.005
66
59
17
17
Table 6.2: Number of iteration steps of the GMRES algorithm, which is needed for different mesh sizes
. The method is applied to
and diffusion coefficients
to reduce the initial residuum by the factor
Example 4.2 for a
partition.
+
1
hyperbolic choice
0.01 0.1 1 10
40
40 46 82
37
38 43 74
19
19 19 18
19
19 19 19
100
197
179
17
19
elliptic choice
0.01 0.1 1 10
40
40 46 82
37
38 43 74
15
15 17 36
15
15 18 36
100
197
179
87
87
Table 6.3: Example 4.2 is considered for a
partition. We see the number of iteration steps of the
GMRES algorithm, which is needed for different constants and diffusion coefficients to reduce the
. The mesh size is chosen by .
initial residuum by the factor
Next, we consider the same constellation for Example 4.2. The flow is given by .
Thus does not vanish on any part of the interface. Therefore the information is transported
across all interfaces. Indeed, in Table 6.2 we observe a different behavior compared to the last
example. In the advection dominated case the number of iteration steps is independent of the
mesh size. Due to the fact, that the solution is dominated by the advection part, especially the
Dirichlet values are exchanged in the direction of the flow. In the diffusion dominated regime we
observe again the dependence on the mesh size .
Dependence on the stabilization parameters
In section 2.6 and section 4.3 we studied the influence of the choice of the stabilization + parameters
3
on the accuracy of the approximation. There, we could validate our choice of the parameters.
Now, by means of Example 4.2 we study the influence of the stabilization parameters and on
the convergence behavior of+ the GMRES
algorithm. Again we use a partition and the same
3 + ).
meshes as in the last paragraph ( First we vary the constant
for
. The results can be seen in Table 6.3. The algorithm
depends only weakly on the parameter . Only in the diffusion dominated case for the elliptic
choice the parameter must3 not be chosen too large; since then mainly Dirichlet values and not
Neumann values are interchanged across the interfaces.
If we vary the parameter
we observe the opposite behavior
(cf. Table 6.4). In the diffusion
3
dominated regime the parameter must not be chosen too small for both parameter strategies. For
small the algorithm is quite insensitive to the choice of .
#
6.3 Numerical results
105
3
1
0.01
190
173
17
19
hyperbolic choice
0.1 1 10
82 46 40
73 43 38
18 19 19
19 19 19
100
40
37
19
19
0.01
190
173
80
80
elliptic choice
0.1 1 10
82 46 40
73 43 38
36 17 15
36 18 15
100
40
37
15
15
Table 6.4: Example 4.2 is considered for a
partition. The number of iteration steps of the GMRES algorithm, which is needed for different constants and diffusion coefficients to reduce the initial
, is printed. The mesh size is chosen by .
residuum by the factor
1
2
25
26
16
6
hyperbolic choice
4
6
8 10
51 65 78 81
21 26 30 34
21 26 30 34
21 26 30 34
12
97
38
38
38
2
25
26
13
13
elliptic choice
4
6
8 10
51 65 78 81
21 26 30 34
19 24 28 33
19 24 28 33
12
97
38
35
35
Table 6.5: Example 4.2 is considered. We print the number of iteration steps of the GMRES algorithm,
which is needed for different diffusion
coefficients to reduce the initial residuum of a factor . Here
the domain is decomposed into
subdomains. The mesh size is always chosen by .
Dependence on the number of subdomains
In this paragraph we study the dependence on the number of subdomains. Therefore we decompose the unit square into sub-squares. Again we use Example 4.2 and the mesh sizes as
above ( ).
As expected we observe an increase of the number of iteration steps for more subdomains in Table
6.5. Furthermore, it is interesting that the number of iteration steps decreases for smaller diffusion
coefficients .
The dependence on the number of subdomains can be reduced by a coarse space. But an appropriate choice of a coarse space is the subject for further research.
#
GMRES method with preconditioner
Now we apply the preconditioner , given by (6.26). As already mentioned the preconditioner
is motivated by the continuous case. The additional stabilization terms of the three-field formulation do not influence the preconditioner. This is precisely the reason, why the performance of the
proposed preconditioner deteriorates, if the the three-field formulation is stabilized. The proposed
preconditioner works well for the diffusion–dominated case in the case of small stabilization parameters. In the advection–dominated case the number of iteration steps for the preconditioned
method is only slightly smaller than in the case without preconditioner.
106
A preconditioned Schur complement method
+
+
3
S TABILIZATION
, +
3
#
,
,
3
P RECOND .
RR
–
RR
–
RR
–
#
#
#
37
30
37
29
47
26
41
62
42
60
62
38
#
46
78
47
75
67
51
#
47
205
51
185
95
72
Table 6.6: is decomposed into squares. We use matching grids with mesh size and the reaction
. The number
coefficient is given by of iteration steps for Example 6.1 is plotted, which is needed
. ’–’ denotes
to reduce the initial residuum by
the Schur complement method without preconditioner;
the preconditioned version is given by ’RR’.
The symmetric case
Let us start with the case, that the bilinear form is symmetric. Thus the preconditioner reduces
to the well known Neumann-Neumann preconditioner. This case is interesting, since in the last
section we have seen that the GMRES algorithm for the stabilized Schur complement equation
without preconditioner converges only slowly
+ in the
3 diffusion dominated regime. In the preconditioned case the conforming theory predicts, that the number of iteration steps depends only weakly
)
on the mesh size. Precisely there holds (for
9 >
(6.31)
is the maximum diameter of the local
(cf. P. L E TALLEC, Y.H. D E ROECK [RT91] ), where
subdomains.
We start our investigation with a simple diffusion-reaction equation.
Example 6.1 Let the right hand side
be chosen in such a way that
?
(6.32)
?
becomes the exact solution of
in
on
#
In order to analyze the effect of the stabilization
we begin with the conforming case: Defining a
global mesh in , the meshes , and are defined by the restriction of on , and
. The meshes are non-regular and uniform. The global mesh size is denoted by .
Since the + inf-sup conditions
(Ass. 3) and (Ass. 4) are fulfilled, the simple discrete three-field
3
formulation (3.17) is also well posed. Hence we can neglect the additional stabilization terms by
choosing
and .
In Table 6.6 we can observe that the GMRES algorithm without preconditioner converges slowly
for fine meshes. In contrast the preconditioned Schur complement equation shows a different
6.3 Numerical results
107
1.2
0
10
1
−2
10
0.8
−4
10
0.6
−6
10
0.4
−8
10
ε =1, P
ε =1,
ε =10−6, P
ε =10−6
0.2
0
−10
10
−12
−0.2
−0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
50
100
150
200
250
300
350
10
Iteration steps
Figure 6.1: On the left hand side the flow of Example 4.1 is plotted. On the right hand side we see
the residuum of the Schur complement equation
for Example 4.1 in dependence on the iteration steps. ’P’
denotes the preconditioned version. We use a partition and the parameters are chosen by ,
.
convergence behavior: the number of iteration steps is almost independent of the mesh size. Quite
different is the situation in the stabilized case. There, both algorithms depend on the mesh size
and the preconditioned method works always worse. In addition, we clearly see the effect of the
stabilization in the case without preconditioner. Due to the stabilization for the number
of iteration steps could be reduced by the factor .
#
The nonsymmetric case
Next we study the effect of the advection with the help of Example 4.1. Remember, that the given
flow is a rotational flow, where vanishes on the whole boundary
(cf. Figure 6.1). In
addition, all characteristics are closed. As already discussed in section 4.3 solving this problem
is extremely difficult for any domain decomposition method. Defining a global mesh size the
local mesh sizes are given by the checkerboard pattern (6.29). The stabilization parameters
in
+
this
section
are
always
given
by
the
elliptic
choice
(6.28).
For
the
Schur
complement
equation
3
+
,
without
preconditioner
the
constants
of
the
stabilization
parameters
are
determined
by
3
+
3
. Due to the results of the symmetric case for the preconditioned method we choose ,
in the diffusion dominated case and , in the advection dominated case.
In order to get an impression of the GMRES algorithm, the residuum in dependence on the iteration
steps is plotted in Figure 6.1. In the advection dominated case we observe a stage, where the
residuum almost stagnates. This behavior is typical for the GMRES method.
Let us consider Table 6.7. First, we observe, that we need much more iteration steps in the advection dominat regime. There, the structure of the problem is quite complicated. The solution
possesses strong layers.
We observe, that the preconditioned method always needs less iteration steps for fine meshes than
the method without preconditioner. But due to the preconditioning the number of iteration steps
is only slightly reduced. Since the computation of the preconditioner is expensive, the application
#
108
A preconditioned Schur complement method
P RECOND .
RR
–
RR
–
RR
–
RR
–
#
32
23
31
24
60
63
61
64
#
#
62
36
36
37
88
112
103
121
#
50
50
46
51
119
157
163
186
38
69
38
72
111
179
209
289
Table 6.7: is decomposed into squares. The meshes are chosen by a checkerboard pattern with
.
mesh sizes (6.29). is a parameter for the global mesh size. The reaction coefficient is given by The number of iteration steps is printed, which is needed to reduce the initial residuum of Example 4.1 by
. ’–’ denotes the Schur complement method without preconditioner; the preconditioned version is
given by ’RR’.
of the preconditioner is only meaningful in the diffusion dominated case for fine meshes.
In the advection dominated case the performance of the preconditioned version deteriorates. This
is caused by the additional -dependent stabilization terms. These terms are not incorporated into
our preconditioner.
Finally, we modify Example 6.1. This time the problem is nonsymmetric:
Example 6.2 Let the right hand side
be chosen in such a way that
?
(6.33)
?
becomes the exact solution of
,
, (b) in
on
#
. This example is also treated by F.C. O
We consider
three cases: (a)
, ,
and (c)
TTO [Ott99]. Again, we use the
above checkerboard pattern for our
meshes.
In
order
to
ensure
the
assumption (Ass. 2a) we choose
the reaction coefficient by . The results are given in Table 6.8. We see the same results as
in the previous cases. In the diffusion dominated case (a) the preconditioned algorithm shows
smaller iteration numbers. Especially for the case of very fine meshes we suggest the application
of the preconditioner.
In the advection-dominated case (b) the preconditioner should be neglected, since the number of
iteration steps is always smaller without preconditioner. In case (c) the application of the preconditioner is only meaningful in the case of very fine meshes.
In addition, in case (c) the number of iteration steps is significantly larger than in case (b). This is
caused by the fact that vanishes on parts of the interface. Thus there is no coupling by the
additional stabilization terms.
6.4 Conclusions
109
(b) , (c) , (a) , P RECOND .
RR
–
RR
–
RR
–
#
33
22
45
28
48
34
#
38
35
50
31
61
50
#
48
48
86
34
106
70
#
40
66
59
34
69
105
Table 6.8: is decomposed into squares. We use the checkerboard pattern (6.29) with global mesh
. The reaction coefficient is given by . The number of iteration steps is plotted,
size . ’–’ denotes the Schur complement
which is needed to reduce the initial residuum of Example 6.2 by
method without preconditioner; the preconditioned version is given by ’RR’.
6.4 Conclusions
Starting from the three-field formulation (3.6) on the continuous level we could derive the Schur
complement equation. The interdependence between both equations was shown. Moreover, we
propose a preconditioner for the Schur complement equation. The convergence of the corresponding Richardson iteration could be derived in the diffusion dominated case.
In a next step this technique was transferred to the discrete case. Starting from the discrete stabilized three-field formulation (4.5) we could propose an adapted Schur complement equation
(6.21), which involves the corresponding stabilization terms. It could be shown that the equation possesses a unique solution. Furthermore, it could be proved that the corresponding local
Steklov-Poincaré operators are invertible. Then the inverses were used for the construction of a
preconditioner of the discrete Schur complement equation (6.21).
Finally, the discrete Schur complement equation was numerically solved with the help of the
GMRES algorithm. The numerical experiments show, that the algorithm works quite robustly. The
application of the preconditioner is only helpful in the diffusion dominated case. The singularly
perturbed case requires a preconditioner, where the additional stabilization terms are worked in
more properly.
Chapter 7
An alternating Schwarz algorithm
In this chapter a further iterative decoupling of the three-field formulation is presented. The resulting algorithm is a modification of an iterative scheme of R. G LOWINSKI and P. L E TALLEC (cf.
[GT89], ch. 3.4), called ALG 3. First they applied the scheme to Augmented Lagrange methods in
nonlinear mechanics. Later they noticed the strong connection to domain decomposition methods
(cf. [GT90]).
Following the line of F.C. OTTO [Ott99] we present the continuous case first. Then the iterative
scheme is discussed for non-conforming and conforming discretizations. Finally we give some
numerical results.
7.1 The continuous formulation
To avoid technical problems, let us assume for this chapter that the decomposition
strip-wise, i.e.
5 H< is
with
. Let be a solution of the continuous three-field
formulation (3.6). Moreover we impose the following regularity assumptions:
H < H< $#
The starting point is the discrete stabilized three-field formulation (4.5) of the advection-diffusion
equation. But instead of the discrete function spaces we will use the continuous function spaces.
In addition, we will neglect the consistent, additional stabilization term in the second line of (4.5),
since in our iterative decoupling strategy this term always vanishes.
=< ( =< =< ! ( #
(7.1)
112
An alternating Schwarz algorithm
( ( ; ( For the additional term we have used+ the notation
+
+
+
% + Q . #
+
In the last chapters the parameter
+
+
(
(
was fixed by
& +
resp.
+
(
#
But on the continuous level there
are no grids to determine . Therefore we replace the parameter
by a global
where is a global parameter representing the mesh size of a
,simplifies
discretization. Then to
( ( +
% + Q . #
Notice, that for this simplified choice there holds
( ( . #
(7.2)
Furthermore we observe that due to the consistency the continuous solution also satisfies
the stabilized three-field formulation (7.1).
Now we adapt the ALG 3 of R. G LOWINSKI
and P. LE TALLEC (cf. [GT89], ch. 3.4) to our
problem. Then for an initial guess
Find such that
Q
Compute
Find
Compute
for all 5 by
Q DE
DE
( #
+ Q $#
D (7.3)
(7.4)
with
; =< (
+ Q the following algorithm is proposed:
,
by
+ Q .
Q ! ( DE Q DE
(
#
D
(7.5)
(7.6)
7.1 The continuous formulation
113
The algorithm can be interpreted as a Richardson iteration of the second line of (7.1). In addition,
in the first line we have neglected the additional streamline diffusion terms, because the algorithm
is given on the continuous level.
Remark 7.1 The solution of the first step (7.3) of this algorithm is given by the solution of the
following local boundary value problems:
(
Q
+
S( in
on
on
(7.7)
( . #
with
Remark 7.2 The above algorithm is well posed, i.e. the unknowns in the equations (7.3), (7.4),
(7.5) and (7.6) are uniquely determined.
This is obvious for (7.4) and (7.6). The well-posedness of (7.5) is a consequence of .
Finally (7.3) possesses a unique solution, because of
0 6 S( 0 " " #
( In a next step it is proved, that the method is equivalent to an alternating Schwarz method with
Robin conditions at the interface. (cf. F.-C. OTTO [Ott99] or R. G LOWINSKI , P. L E TALLEC
[GT90]):
###
Theorem 7.1 Assuming (7.2) the sequence of solutions
) :
following iterative scheme for Q
(
# # # ( can be obtained by the
#
on in
on
(7.8)
Proof: Since the partition is strip-wise we can write
;
Q
+ Q $#
Therefore the algorithm can be rewritten in terms living on the intersections
for (7.4), (7.5) and (7.6):
Q Q
( Q ( . Thus we obtain
(7.9)
(
Q
(
S(
(
(
(
(7.10)
(7.11)
114
An alternating Schwarz algorithm
+ Q . Using the last representation of the algorithm by simple computations we
( F D Q S ( ( ( for all
arrive at
!
!
!
of and . Using the identity
(7.10) yields
( ! ( ( ( ! Q S ( ( ( ( Q ! ( ( ( ! ( D F ( ( ( ! (
L ! ( U #
on the intersection
or
In a next step we use the differential interpretation (7.7) of step (7.3) and insert the boundary
:
condition on
! ( (
(
L
U
! ! ( ! ( ! ( ( (
>
9
! ( ( ( ( ( ( # (7.12)
!
!
Using assumption
(7.2) and again the representation (7.7) we obtain the assertion by
!
(
( ( #
Remark 7.3 Unfortunately in the discrete case for nonmatching grids the assumption (7.2) is in
gives
general not fulfilled. But then the equation (7.12) on the interface
(
! S
(
(
(
! !
#
7.1 The continuous formulation
115
So we observe that the transmission condition is almost the same. It differs only by a small additional term caused by the nonmatching grids.
This alternating Schwarz algorithm given in the form (7.8) is thoroughly discussed by G. L UBE
and coworkers (cf. [LMO00, OL99]) or F. NATAF and coworkers (cf. [NR95, Nat99, JNR00]).
Because of the fact, that the algorithm interchanges Robin interface conditions, the algorithm is
sometimes called Robin-Robin algorithm.
S ( on for our choice ensures that the sequence
Remark 7.4 The property converges
of (3.6) if we impose (7.2) and that the initial solution
to the solution fulfills for ) . Proofs can be found in the works cited above. All proofs
are based on a technique of P.L. L IONS [Lio90]. Unfortunately in general the convergence is not
linear (cf. F.-C. OTTO ([Ott99], Theorem 3.4).
### Remark 7.5 G. L UBE and coworkers (cf. [OL99, LMO00]) also derive an a posteriori estimate
for the alternating Schwarz algorithm and extend the results to parabolic equations (cf. [LOM98])
and to the Oseen equations (cf. [OLM01]).
(
The choice of the acceleration parameter is very interesting. Practical experiments show
that the Robin-Robin algorithm is very sensitive with respect to the choice of the acceleration
parameter. Therefore in the last years great efforts have been made to find a mathematical based,
appropriate choice.
In [OL99, LMO00] a proposal is derived by an a posteriori estimate. Making some simplifying
assumptions by ’equilibration’ of the terms of the a posteriori estimate they obtain
A ( L " " ACED F " N" A CED F U #
( (7.13)
In the first works of F. NATAF (cf. [NR95]) regarding an appropriate choice of the parameter
was chosen in such a way, that the first Fourier mode vanishes. For the analysis they assume
are given by infinite strips. This
constant coefficients and the domain and the subdomains
approach yields
(
#
(7.14)
The result is very similar to the results we get with help of a simple Fourier analysis for bounded
domains (cf. chapter 8). In recent works of F. NATAF [JNR00] also tangential and higher derivatives are considered. The new choice is then determined by minimizing the error over a certain
range of Fourier modes.
+
Our choice in this work
.
(7.15)
S( SA ( is a natural result of our a priori estimate of the stabilized, discretethree-field
formulation. There .
fore it is quite interesting that our choice is nearly the same as S( Remark 7.6 If we choose the acceleration parameter as
9
(7.16)
we get the the adaptive Robin-Neumann method (ARN-method) of A. Q UARTERONI, F. G ASTALDI
(cf. [QV99, GGQ96, ATV98]).
116
An alternating Schwarz algorithm
7.2 Discretization
Conforming case
In this context the conforming case is given, when all the meshes
belonging to the function spaces
are constructed by one mesh of . Then , and are given by the restriction of . Furthermore the polynomial degree of the ansatz functions has to be the
to , resp.
same for the different finite element spaces. This case is discussed in detail by the group of
G. L UBE [LMO00, OL99]. Further results can be found in some articles of F. NATAF (cf. [NR95],
[JNR00]). For an implementation it is very interesting, that the Robin-Robin algorithm can also
be written as a simple fixed point iteration of a corresponding interface equation. The resulting
interface equation can be solved by a Krylov method, like GMRES. This way it is possible to
obtain better convergence results. Details and numerical experiments can be found in F. NATAF,
F. ROGIER , E. DE S TURLER [NRdS95].
Nonconforming case
P. L E TALLEC and T. S ASSI [TS95] describe a non-conforming discretization for the Poisson
problem. They use a technique, which is very similar to the Mortar technique. Here we extend the
algorithm to the advection-diffusion problem. We simply replace the continuous functions of the
last algorithm with the corresponding discrete functions. Starting with an initial guess ,
, the algorithm reads
such that
( Find #
Q by
Q S( #
with
Find ; Q ( H< by
Compute (
Q for all .
Compute
D
(7.17)
D
D
"
(7.18)
(7.19)
D
4
D
D
(7.20)
7.3 Numerical results
117
Again, following the reasoning of Remark 7.2, the algorithm is well posed. A convergence proof
of this algorithm is still an open problem and will be the subject for future work.
7.3 Numerical results
In this section some numerical results are presented. Starting with some remarks about the implementation, our main issue is the comparison of our nonconforming approach with the conforming
approach of G. L UBE and coworkers.
Some remarks about the implementation
Comparing the conforming case and the nonconforming approach the latter one is more involved
since some additional operations are necessary to shift the values from one mesh to a different
one.
But let us discuss the four steps of the algorithm in detail. The realization of (7.17) is straightforward and has already been discussed in the last chapters. Updating
the Lagrange multipliers in
projections of
step (7.18) and (7.20) is more complicated. Here we compute the
( ( resp.
#
Of course for large Lagrange multiplier spaces these steps can be quite expensive. In the conforming case no projection is necessary. But fortunately these computations are only local and
therefore the steps (7.17), (7.18) and (7.20) can be completely done in parallel.
The information between the interfaces is exchanged in step (7.19). There in each iteration step a
global interface problem has to be solved. In our implementation this step is quite expensive, since
each time a linear system must be solved. But in an efficient implementation solving the global
problem can be circumvented by a thorough study of the local components.
Results
In order to compare our results with the conforming
case we consider the following example of
(cf. Example 6.2):
F.-C. OTTO [Ott99] for the unit square
Example 7.1 Let the right hand side
be chosen in such a way that
?
(7.21)
?
becomes the exact solution of
in
#
,
(b)
,
and (c) , .
For the numerical experiments is decomposed into rectangles. The interior meshes
# # have the mesh sizes , the mesh sizes of the Lagrange multiplier spaces are given
We analyze the following three cases: (a)
, on
118
An alternating Schwarz algorithm
Iteration step k=1
Iteration step k=3
1
1
0.5
0.5
0
1
1
0.5
0 0
Iteration step k=5
0
1
0 0
Iteration step k=6
1
1
0.5
0.5
0
1
1
0.5
0 0
Iteration step k=8
0 0
Iteration step k=10
1
0.5
0.5
1
0
1
0.5
1
0.5
0.5
0
1
0.5
0
1
1
0.5
0.5
0.5
0
1
1
0.5
0.5
0
0
0.5
0
Figure 7.1: The absolute
error of Example7.1
(a)
for). different iteration steps is plotted. The domain is
decomposed into rectangles ( Iteration step k=1
Iteration step k=2
1
1
0.5
0.5
0
1
1
0.5
0 0
Iteration step k=3
0
1
0 0
Iteration step k=4
1
1
0.5
0.5
0
1
1
0.5
0 0
Iteration step k=5
0 0
Iteration step k=6
1
0.5
0.5
1
0
0.5
0
1
0.5
1
0.5
0.5
0
1
0.5
0
1
1
0.5
0.5
0.5
0
1
1
0.5
0
0.5
0
Figure 7.2: The absolute
the first iteration steps is plotted. The domain is
error of Example7.1
(b)
for
decomposed into rectangles ( ).
7.3 Numerical results
119
Iteration step k=1
Iteration step k=2
1.5
1.5
1
1
0.5
0.5
0
1
0
1
1
0.5
1
0.5
0.5
0 0
Iteration step k=3
0 0
Iteration step k=4
1.5
0.5
1.5
1
1
0.5
0.5
0
1
0
1
1
0.5
1
0.5
0.5
0 0
Iteration step k=5
0 0
Iteration step k=6
1.5
0.5
1.5
1
1
0.5
0.5
0
1
0
1
1
0.5
0
1
0.5
0.5
0.5
0
0
0
Figure 7.3: The absolute
the first iteration steps is plotted. The domain is
error of Example 7.1
(c)
for
decomposed into rectangles ( ).
1
1
10
10
2
L error
energy error
|φ
−φ | /|φ
k+1
0
10
k
2
|
k+1 2
k+1
0
10
−1
k
2
|
k+1 2
−1
10
10
−2
−2
10
10
−3
−3
10
10
−4
10
2
L error
energy error
|φ
−φ | /|φ
−4
0
10
20
30
iteration step
40
50
60
10
0
10
20
(a)
30
iteration step
(b)
40
50
60
Figure 7.4: The error in dependence on the iteration steps for Example 7.1 in the case of
subdo ) and in (b) the advection dominated
mains. In (a) the diffusion dominated case
is
considered
(
case
). ( , )
( by and the global interface space possesses the mesh size parameter is chosen by
( #
# .
The acceleration
120
An alternating Schwarz algorithm
1
10
1
L2 error
energy error
|φ
−φ | /|φ
k+1
0
10
k
2
0.9
|
k+1 2
0.8
0.7
−1
10
0.6
0.5
−2
10
0.4
0.3
−3
10
0.2
0.1
−4
10
0
10
20
30
iteration step
40
50
60
0
0
0.1
0.2
Mesh for
(c)
0.3
0.4
0.5
0.7
0.8
0.9
1
partition ( Figure
on the iteration steps for Example 7.1 (c) ( 7.5: The error in
dependence
) in the case of subdomains. On the right hand side the interior meshes
plotted.
A ( # #
)
, are
0.6
Up to the term this choice agrees with the choice . In the Figures 7.1, 7.2 and 7.3 the
absolute error for the three different cases in dependence on the iteration step is plotted. All
, . In case (a) the
computations of this paragraph start with the initial guess solution is mainly influenced by the diffusion part (cf. Figure 7.1). The correct solution propagates
isotropically from the boundary into the interior of the domain. In case (b) and (c) the advection
part dominates. And indeed in both cases it can be seen that the solution is propagated in the
shows very fast
direction of the flow . Even the case where vanishes on some interfaces
convergence. The results agree with the conforming case presented in F.-C. OTTO [Ott99].
In Figure 7.4 and Figure 7.5 the convergence of the three cases is considered.
But this time we
, have chosen finer meshes ( , ). The error and the energy
error is plotted. The third error indicator is defined as follows. Denoting the degrees of freedom
of by , we define
#
#
" " "
where "2" is the euclidian norm of the , #
"
. We observe that the
error and
the energy error decrease very fast. This coincides with the conforming results of F.-C. OTTO.
Moreover we see, that the third error indicator coincides with the error until the discretization level
as a convergence criterion for our algorithm.
is reached. Therefore we use
In a next step we consider the different choices of the acceleration parameter in dependence on
the mesh size. Again the unit square is decomposed into
rectangles. For a global mesh
meshes are given by a checkerboard pattern with local mesh sizes ,
size the local
and . We analyze the choice (7.13) of G. L UBE ET AL ., the elliptic choice
and the hyperbolic choice of the three field formulation and the choices of A. Q UARTERONI ET
AL . (cf. (7.16)) and F. NATAF ET AL . (cf. (7.14)). Since the reaction term is zero for our
example, the parameter strategies of F. NATAF ET AL . and A. Q UARTERONI ET AL . coincide. In
Table 7.1 we see the number of iteration steps which are needed to achieve the accuracy for different mesh sizes . If the convergence is not reached within 500 steps, we denote this by a
7.3 Numerical results
Case
(a)
h
0.05
0.02
0.01
0.005
0.05
0.02
0.01
0.005
0.05
0.02
0.01
0.005
(b)
(c)
121
Lube
90
98
108
91
10
10
10
9
10
11
12
13
hyperbolic
64
131
231
403
10
10
10
10
7
8
12
17
elliptic
64
131
231
403
58
57
57
57
103
156
220
279
Nataf
317
420
–
–
10
10
9
9
Table 7.1: We consider Example 7.1 and analyze different parameter strategies in dependence on the mesh
, is printed. is decomposed
size . The number of iteration steps, which is needed to achieve
into rectangles.
Lube
8
11
14
17
20
23
hyperbolic
8
11
14
17
20
23
elliptic
42
58
72
86
100
114
Nataf
8
10
13
16
19
22
Table 7.2: We consider Example 7.1 (b) and analyze different parameter strategies in dependence on the
, is printed.
number of subdomains.
The number of iteration steps, which is needed toachieve
is decomposed into
subdomains and the mesh size is always .
dash ’–’. The proposal of A. Q UARTERONI ET AL . and F. NATAF ET AL . does not work in case
(c), since the linear system in (7.19) is singular.
In nearly all cases the proposal of G. L UBE ET AL . works best. Moreover we observe, that in
the singularly perturbed cases (b) and (c) the algorithm works quite effective. Only the elliptic
choice shows a bad performance. In addition, one can observe that the number of iteration steps
for the choice of G. L UBE ET AL . and A. Q UARTERONI ET AL . is independent on the mesh
size. If we use the elliptic or the hyperbolic choice in the diffusion dominated case (a), we observe
that the algorithm deteriorates for finer meshes. Summarized one can say that the acceleration
parameter should be chosen independent of the mesh size but should take into account the diffusion
coefficient . Moreover, the proposed algorithm works much better for than in the diffusion
dominated regime.
The same results can be seen in Table 7.2. There for Example 7.1 (b) the number of subdomains
). It can be observed that the number of iteration steps
is increased for a fixed mesh size ( increases slightly for a larger number of subdomains.
#
122
An alternating Schwarz algorithm
7.4 Conclusions
In this chapter we have shown how the classical alternating Schwarz methods can be derived from
the stabilized three field formulation. With the help of different parameter choices we could generalize well-known algorithms to the nonconforming case. In contrast to the approach of P. L E TAL LEC and T. S ASSI the choice of the discrete spaces is almost arbitrary. Furthermore the scheme
is applied to the nonsymmetric case. Finally numerical experiments confirm our approach. In the
advection dominated case we obtain very promising results, which are competitive to all known
algorithms. In the diffusion dominated regime the algorithm shows a quite bad performance. But
maybe the performance can be improved by an adaptive choice of the acceleration parameter.
Chapter 8
Comparison of some nonoverlapping
domain decomposition methods
In this chapter the performance of the nonoverlapping domain decomposition methods of chapter
6 and chapter 7 are compared. First a simple Fourier analysis is performed for both algorithms. It
will be seen that the Schur complement method of chapter 6 possesses much better convergence
properties than the alternating Schwarz algorithm of chapter 7. In parts this will also be verified
by some numerical results, which will be presented afterwards.
8.1 Fourier analysis
We start the analysis by considering a simple model case with constant coefficients. Thus we can
perform a Fourier analysis for both cases. We will observe that the Schur complement algorithm
converges linearly in contrast to the alternating Schwarz algorithm.
The preconditioned Schur complement method
The Fourier analysis will be discussed in one dimension and in two dimensions on the basis of two
subdomains. The one-dimensional case can be seen as a motivation. The two-dimensional case
is more interesting, because the convergence properties of many domain decomposition methods
strongly depend on the direction of . In one dimension the direction is already determined by the
sign of .
.
Fourier analysis for the one-dimensional case
The one-dimensional problem is given by
.
. . . . #
in
We assume that the coefficients
, and are constants. Additionally, we require
. Restricting ourselves to the case of two subdomains, we decompose the domain
into the subdomains and
with (cf. Figure 8.1).
The interface is given by .
.
)
M 124
Comparison of some nonoverlapping domain decomposition methods
Ω1
Ω2
0
l
L
Figure 8.1: The one-dimensional test case: The subdomains are given by
and
.
We consider the preconditioned Richardson iteration (6.15)
Find
(8.1)
. By virtue of
of the Schur complement equation (6.3) with preconditioner in the -th iteration step can be derived. denotes the
(8.1) an equation for the error
solution of the Schur complement equation (6.3). Then the Richardson iteration can equivalently
be written as
or
#
(8.2)
In chapter 6 the following differential form of the algorithm was derived for . L . U
:
in
on
on (8.3)
. on #
in
on
(8.4)
Finally the new interface error is given by
L < < U #
Remark 8.1 Note that the solution of (8.3) is the error between the solution of the
and the true solution in .
(8.5)
-th step
Now we use the differential form in order to derive an explicit formula for the error. Let us start
with (8.3). The general solution of the equation
is given by
with
% -
.
(
.
-
#
8.1 Fourier analysis
125
and - 0 - with appropriate constants 0 . Analogously for the second equation (8.4) we have
- - 0 Using the boundary conditions at
?
with constants
?
, we obtain
?
?
. Inserting the interface
conditions on of (8.3) we obtain the solutions
- ? - ? $#
?
?
" & . < the interface condition on of (8.4) can be reformulated by
" " for and . Calculating
-1
-1
" " Defining
?
?
:
and
?
we obtain for
?
?
?
-1
-1
and
-1
#
This yields
< #
<
Analogously we derive for :
< #
Inserting this into (8.5) yields
< < ?
?
?
?
?
?
?
4
with
#
(8.6)
can be interpreted as the contraction rate of our algorithm. It gives the decay of the error from
one iteration step to the next one. Hence we obtain
126
Comparison of some nonoverlapping domain decomposition methods
0
0
10
|KDR|
|K|
10
−5
10
L=1.1
L=3
L=10
L=100
−10
10
−5
10
0
0.2
0.4
0.6
ε
0.8
L=1.1
L=3
L=10
L=100
−10
10
1
0
0.2
0.4
(a)
, (Robin-Robin preconditioner)
ε
0.6
0.8
1
#
(b)
, , (Dirichlet-Robin algorithm)
Figure 8.2: The absolute contraction rates in dependence on the diffusion coefficient for different subdo, , )
main widths. ( Theorem 8.1 Suppose
#
. Then the Richardson iteration (8.1) converges if and only if
Thus we get the following condition for the relaxation parameter :
Corollary 8.1 Let
be given. Then the Richardson iteration converges if and only if
#
to achieve optimal
But how should we choose the relaxation parameter for given
convergence? In the one-dimensional case this is quite simple. If we choose
and therefore the algorithm converges in one step. Unfortunately we will see, that
we get
such a choice is not possible in more than one dimension.
Remark 8.2 If we choose
contraction rate reduces to
and
we obtain the Dirichlet-Robin algorithm. Then the
$#
This result agrees with the work of A. A LONSO ET AL . ( [ATV98], section 2), A. Q UARTERONI ,
A. VALLI ([QV99], ch. 6.2 ) or F. G ASTALDI ET AL . ( [GGQ96], section 3).
In Figure 8.2 the contraction rate is plotted for the two most important parameter choices. On
the left hand side the contraction rate of the Robin-Robin preconditioner is plotted; on the right
hand side we see the Dirichlet-Robin algorithm. The plots show that the algorithms are sensitive to the width of the subdomains. For very narrow subdomains the convergence deteriorates.
Moreover, the algorithms converge better in the advection dominated case. This agrees with our
numerical experiments.
8.1 Fourier analysis
127
y
Γ
Ω
Ω
Ω
2
1
Α
Figure 8.3: The domain
x
L
in the two-dimensional case.
Fourier analysis for the two-dimensional case
Now we generalize the Fourier analysis to the two-dimensional case. We set
and consider the following boundary value problem
Again, we decompose into
% . Furthermore, the coefficients
(8.2), we obtain the differential form:
"
Inserting the ansatz
into (8.7) we arrive at
..
#
in
on
on
(8.7)
(8.8)
. *%
. Finally the interface error is updated by
in
on
" *%
in
on
on and
*% with interface
and are constants. Starting again with equation
" with
and L <
< U #
.. !
(8.9)
128
Comparison of some nonoverlapping domain decomposition methods
?
The resulting ordinary differential equation for = ( 4H ( 4
% (
0 ( 4= (
K(
with constants 0 and 0 for all )
?
(8.7) is given by
into the general solution of
5 0 K ( 4 we obtain
(8.10)
(8.10) given by
imply 4
and eigenvalues :
is given by
#
..
and Inserting the boundary conditions # # # #
with a constant independent of . The boundary conditions for
. Solving the arising eigenvalue problem we obtain a set of solutions
?
. Combining the results we see that the solution of
; < 0 ( 4H ;
< 0 K( 4 $#
?
?
?
?
It can be seen that the series and its derivatives converge uniformly if the initial guess is sufficiently
regular. The use of the boundary conditions on yields
* % * % ; < 0 ( 4 ;
K( 4 0
<
*%
%
?
*%
(8.11)
?
%
?
(8.12)
?
. Hence we can infer
0 ( 4 0 K( 4 for all . Analogously for the solution of (8.8) we can derive
;
< ( 4= K( $#
;
< 4 for all %
?
%
?
?
?
?
?
(8.13)
8.1 Fourier analysis
From
"
129
" " K ( ?
*%
?
%
?
%
?
%
?
?
we obtain
4 4 0 ( 4 0 K( 4 4 0 ( 4 0 K( #
%
%
*%
and the boundary conditions on
( *%
?
" *%
?
; < 4 ( 4 ;
< 4 K( 4 ;
< 4S0 ( 4 ;
K( 4 4
S
0
<
%
?
?
%
%
%
?
?
%
0 ( 4 L
0 ( 4 4 4 %
?
%
?
?
%
?
%
?
?
4 % ( 4 %
?
K ( U
4 0 ( 4 0 K( 4
4 0 ( 4 0 K( #
%
%
?
%
(8.15)
?
Now inserting (8.14), (8.15), (8.11) and (8.12) into (8.9) we derive
0 ( 4 (8.14)
?
%
?
%
?
%
?
The last equation is transformed into
0 ( 0 ( 0 ( 4 %
4 0 K( 4
4 0 ( 4 4 %
?
with
%
%
?
( 0
+
%
With the help of (8.13) we can conclude
for all ?
4
%
0 K ( #
%
?
S0 ( 4 %
(8.16)
4 4 %
%
$#
Remark 8.3 The convergence of the preconditioned Schur complement algorithm is determined
by the relation
(8.16). It indicates
the reduction of the error for the different Fourier modes.
Due to the contraction rate depends on the Fourier mode and the
coefficients , and . It is very surprising that the convergence of the algorithm does not depend
on the direction of the flow . Instead, the algorithm is only influenced by the modulus of .
130
Comparison of some nonoverlapping domain decomposition methods
Our next goal is a convergence proof for the Richardson iteration in the two-dimensional case. To
this end we first study some properties of the function
with constants
Lemma 8.1 $
)
( $ *)
.
( satisfies
( $ ( % Furthermore, there holds ) .
$ )
for
)
( #
$ and ( for
Proof: (i) The rule of de l’Hospital yields
$ (
$
$ #
$ ( .
(ii) Due to it is obvious that and . We obtain
(iii) Suppose $ $ (
D F F
D
% . The case $ is treated analogously.
because of
(
K(
It is sufficient to consider the constants 0 , because the constants 0 are coupled with the
(
constants 0 by equation (8.13). Moreover, it is obvious, that the algorithm can only converge if
for all .
. Then the Richardson iteration can only converge if
Lemma 8.2 Assume 2(
(8.17)
A 4 A *( 4 for all .
*)
*)
?
)
)
?
)
*)
)
Now we have to show that it is possible to choose
last lemma is satisfied for all .
Theorem 8.2 If we choose
in the
-norm.
in such a way, that the condition (8.17) of the
sufficiently small, the Richardson iteration converges linearly
8.1 Fourier analysis
131
. Then Lemma 8.1 implies, that
*( A 4 A 2( 4 % there exists such that for all *( A 4 % #
Proof: Without loss of generality we assume %
%
and for given
Thus we can estimate
for all
2( A 4 2( A 4 . Now we choose
Then we can find a constant
A *( 4 in such a way that
0
< ( with
% 0
(8.18)
for all
.
Next we consider
the convergence speed. The error
will bemeasured in a weighted
-norm. We define the weighted -norm in
by
a weighted
#
*(
A 4
-norm and
which is equivalent to the usual -norm. The weighted -norm is then given by
( ( ( ( " '" " 1" ( ( ( ( ( (
" 1" ( ;
< 0 4 ;
< 0 K( 4 also being equivalent to the usual one. Using that the -th step error on the domain
?
?
?
?
and taking into account the relations
?
?
?
?
?
?
?
?
is given by
132
Comparison of some nonoverlapping domain decomposition methods
;
" " ( ( ; L 0
<
" " ( ( ; L 0
we obtain for the error norms at iteration step
"
" "
where
3
A ? 3 ?
. . <
( U
K ( U
3
3
3
L 0 ( U <
; K
(
< L 0 U
" ( ( ( (
.
3
?
?
A . #
for all we can conclude
Owing to (8.18) and the fact that , , , for and " " ( ( % 0 " " ( ( #
3
3
?
?
Remembering that the weighted norms are equivalent to the usual ones the proof is complete.
Now we illustrate the contraction rates. On the left hand side of Figure 8.4 the contraction
rates in dependence on the number of the Fourier mode are shown, where we have chosen and . We can observe,
, ,% , , , that the reduction of the error from step to step is extremely large. Additionally the contraction
rates become smaller for larger Fourier modes and smaller viscosity . The right hand side of
Figure 8.4, where the dependence on the viscosity is presented for different Fourier modes ,
shows the same result.
The only drawback of the algorithm is the dependence on the width of the subdomains. We
choose again , and all other parameters like above and modify % . For small %
the subdomain
% becomes small. In Figure 8.5 we observe, that the contraction
.
rate grows for %
In a nutshell one can say that the preconditioned Richardson iteration converges very fast for nearly
all possible combinations of the parameters. But it is obvious that the convergence speed is very
and . In the case of more than two subdomains a
sensitive with respect to the parameters ,
and is not trivial.
good choice of the parameters
Very interesting is the question, what happens to the contraction rate in the case of many subdomains. The work of Y. ACHDOU ET AL . [ATNV00] treats this in the case of ) infinite strips.
# # # 8.1 Fourier analysis
133
0
0
10
10
ε=1
ε=0.5
ε=0.1
ε=0.05
−5
−5
10
|K|
|K|
10
−10
−10
10
10
l=1
l=2
l=3
l=5
−15
−15
10
10
1
2
3
4
5
0
6
0.2
0.4
l
ε
0.6
0.8
1
Figure 8.4: Contraction rates of the Fourier modes of the preconditioned Richardson iteration.
, , , , , )
( 5
5
10
10
0
0
10
−5
|K|
|K|
10
10
l=1
l=2
l=3
l=5
−10
10
−5
10
−10
10
−15
10
l=1
l=2
l=3
l=5
−15
−4
10
−2
10
0
10
10
−4
10
−2
0
10
# Figure 8.5: Contraction rates in dependence on the subdomain width. ( , )
A
10
A
,
,
,
With the help of the Fourier analysis they obtain the following result for the GMRES algorithm:
The algorithm stagnates during the first ) iterations and then converges rapidly.
The Robin-Robin algorithm
We compare the preconditioned Schur complement algorithm with the Robin-Robin algorithm,
which can be found in the work of G. L UBE ET AL . (cf. [LMO00], [OL99], [Ott99]). As already
explained the algorithm strongly depends on the direction of the flow . Therefore we neglect the
discussion of the one-dimensional case. But a detailed discussion can be found in [Ott99].
Let us introduce the Robin-Robin algorithm for the two-domain problem from above. At iteration
134
Comparison of some nonoverlapping domain decomposition methods
. . . . . G. L UBE
ET AL .
A. Q UARTERONI
F. NATAF
ET AL .
ET AL .
hyperbolic choice (4.28)
elliptic choice (4.30)
Table 8.1: Different choices of the acceleration parameter.
step
the algorithm with relaxation is given by
and
the error
with formulas
in
on
on
in
on
on
#
are free parameters. With an analogous Fourier analysis it can be shown, that
of the -th step can be represented by
&
where
; ( 4= <
; K( 4 <
?
?
?
?
. Inserting the boundary conditions at , we can derive the recursion
$
( K ( . . ? 4 % 4 ? 4 % 4 % 4 4 % ? 4 %
4 ? 4 % #
K ( ( . . 4? % 4 ? 4 % ?
?
Next we study the effect of the parameters on the convergence behavior. With the ansatz
( K( . . we can receive the parameter choices of chapter 7, which are summarized in Table 8.1. Here is
a parameter for the global mesh size of any discretization. If we additionally require we can
derive
(8.19)
( (
8.1 Fourier analysis
135
ε=1
0
ε=0.01
0
10
10
−2
−2
l
|Kl|
10
|K |
10
Nataf
Lube
Quarteroni
3F−ellipt
3F−hyperb
−4
10
0
1
10
2
10
−4
10
3
10
Nataf
Lube
Quarteroni
3F−ellipt
3F−hyperb
0
10
1
10
2
10
3
10
l
10
l
Figure 8.6: The contraction rates of the Fourier modes of the Robin-Robin
for different choices
, , , , algorithm
of the acceleration parameter. ( , )
where
for 4 4 4 4 4 4 4 4 4 4 4 4 or
4 4 4 4 #
4 4 4 4 Thus we have for each Fourier mode the contraction rate .
,
In Figure 8.6 we illustrate the contraction rates for different with given constants
#
, , . The
# acceleration parameters are chosen by Table 8.1 and
the mesh size is determined by . We observe, that the contraction rates tend to for
%
?
%
?
%
?
?
%
%
?
%
%
?
%
?
%
%
?
%
%
%
different and all parameter choices. This means that the small Fourier modes with high energy
are damped rapidly; for larger Fourier modes the convergence deteriorates. Moreover, it can be
clearly seen, that for smaller the contraction rates increase weaker. The comparison with the
results of the Robin-Robin preconditioner of the Richardson iteration shows, that the contraction
rates are significantly larger. But the main difference of the two methods is, that in the case of the
Robin-Robin algorithm the contraction rate increases in dependence of the Fourier mode , while
in the case of the preconditioned Richardson iteration the contraction rate decreases. This explains
why the second method converges linearly and the first one does not converge linearly.
Now we analyze the performance of the acceleration strategies in Figure 8.6. In the diffusion
dominant case ( ) we observe that the choice of G. L UBE ET AL . and the choices, which are
derived by the three-field formulation, damps the lower modes better than the other choices. For
higher modes there is almost no difference between the various suggestions. The same behavior
). This time the elliptic choice
can be also observed in the advection-dominated case ( of the three-field formulation possesses for low modes significantly larger contraction rates than
the remaining ones. Again, among all choices the suggestion of G.+ L UBE + ET AL . gives the best
results.
In Figure 8.7 the dependence on the direction of the flow +
is considered. In
?
the last figure we have seen that the hyperbolic choice gives better
results? than the elliptic choice.
Therefore we analyze only the four remaining choices. For the flow is parallel to the
#
136
Comparison of some nonoverlapping domain decomposition methods
ε=0.01
Nataf
Lube
Quarteroni
3F−hyperbolic
1
b
0.8
|Kl|
α
0.6
0.4
direction of 0.2
0
0
pi/8
pi/4
α
3/8 pi
pi/2
Figure 8.7: On the right hand side the contraction rates for the 5th Fourier mode of the Robin-Robin
, , , algorithm
in the dependence on the angle . ( , are
plotted
, , )
interface. As expected we observe, that the choice of A. Q UARTERONI ET AL . does not converge.
We get the best convergence results, if the flow is orthogonal to the interface. Again we see, that
the proposal of G. L UBE ET AL . gives the best results.
:
Finally we present a lemma which shows the asymptotic behavior for Lemma 8.3 For the contraction rate
precisely for the elliptic choice we obtain
and for all other choices
does not converge to
.
.
. .
. . . . #
for all
. More
Thus in the limit the algorithm does not converge in the case of a flow , which is parallel to the
interface.
8.2 Numerical results
In this thesis we deal with two different classes of domain decomposition methods. In chapter 6 we
discussed a Schur complement method, which was derived by eliminating the interior degrees of
freedom. Moreover, we proposed a preconditioner for the Schur complement equation and solved
the corresponding equation with the help of the GMRES method.
In chapter 7 we presented a second algorithm, the alternating Schwarz method. There, we transferred the well-known case of conforming meshes to the non-conforming case.
8.2 Numerical results
#
Case
(a)
#
#
#
#
(b)
#
#
#
#
(c)
#
#
#
137
method
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
%
5
4
6
7
6
6
9
6
6
13
7
6
6
4
5
6
4
5
6
3
5
6
3
5
9
5
4
17
4
5
23
3
5
25
3
5
%
%
error
–
–
–
12
6
9
15
7
9
21
7
9
–
–
–
8
7
7
7
5
6
7
5
6
–
–
–
22
8
–
31
6
7
38
5
8
–
–
–
–
–
–
–
11
51
25
19
13
–
–
–
–
–
–
–
–
–
9
8
8
–
–
–
–
–
–
–
–
–
51
8
–
%
%
energy error
4
5
6
7
6
6
9
6
5
13
7
5
4
2
4
4
3
4
4
3
4
4
2
4
7
3
3
9
3
3
11
3
3
10
3
3
–
–
–
–
–
–
16
10
56
20
9
24
–
4
6
6
4
5
6
3
5
6
3
5
–
5
4
17
4
5
25
4
5
29
4
6
%
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
9
8
7
7
7
7
7
7
6
–
–
–
23
9
–
33
7
8
43
7
–
Table 8.2: We consider Example 6.2. is decomposed into squares. Again, we use the checkerboard
. The number of iteration steps is plotted,
pattern (6.29). The reaction coefficient is given by which is needed to achieve certain error bounds. The case, that the error bound is not reached within 100
steps, is denoted by ’–’. We use the following notation: ’S’ denotes the Schur complement method without preconditioner, the preconditioned Schur complement method is denoted by ’RR’, and the alternating
Schwarz method is given by ’AS’.
138
Comparison of some nonoverlapping domain decomposition methods
Now we compare both methods with the help of some numerical studies. Unfortunately, a direct
comparison of the needed iteration steps is not possible, since we used different stopping criteria
for the methods. For the Schur complement method we used the reduction of the residuum. In
contrast the convergence of the alternating Schwarz method was controlled by the variation of the
interface variable.
Therefore, within this section we propose the following common convergence criterion. We count
the number of iteration steps, which is needed to achieve certain error bounds.
Here we measure
the deviation of our approximation
from
norm and in the
a given reference solution in the
energy norm .
In order to compare the algorithms, it is necessary to estimate the cost of one iteration step for the
different methods. The Schur complement equation without preconditioner requires the solution
of a local Dirichlet problem in each subdomain. In addition, the Krylov basis must be updated. For
a large number of iteration steps, the algorithm needs a large amount of memory, since the whole
basis of the Krylov space must be stored. The cost of the preconditioned Schur complement
method is almost doubled, since, additionally, local Robin problems have to be solved in each
iteration step. Also the alternating Schwarz method requires the solution of local Robin problems.
Additionally, the solutions of the local Robin problems have to be transferred from one grid to
another in step 2 and step 4. Step 3 requires the solution of an interface problem. Since the
solutions of local problems are the most costly steps in all methods, one iteration step of the Schur
complement method without preconditioner and one step of the alternating Schwarz method are
clearly cheaper than the application of the preconditioned method.
+
3
Again we consider Example 6.2. For the Schur
complement
method we use the elliptic choice
+
3
, in all computations.
Only in the
of the stabilization parameters with constants
+
preconditioned case for case (a) we choose
, . In the case of the alternating
Schwarz method we always use the choice of G. L UBE for the parameter
within this section.
In Table 8.2 the number of iteration steps
which is needed to achieve an error of the
is printed,
norm which is smaller
than , , resp. . The error bounds for the energy
error are given by ,
, resp. . The domain is decomposed into we distinguish
squares and the discretization is given by the checkerboard
pattern (6.29).
Again,
and (c)
the following
three
cases:
(a)
,
,
(b)
,
" " ( " #" ( #
, . The reaction coefficient is always given by .
In the diffusion dominated case (a) we observe, that the preconditioned Schur complement method
and the alternating Schwarz algorithm are independent of the mesh size. As predicted in chapter
6 the Schur complement method without preconditioner deteriorates for finer meshes. In my
opinion the most effective method in the diffusion dominated case is the preconditioned Schur
complement, although two local boundary problems have to be solved in each iteration step.
The situation differs in the advection dominated cases. In case (b) the term does not vanish
anywhere. It can be observed, that all three methods work similarly. Therefore I recommend the
cheap alternating Schwarz method.
In case (c) vanishes on parts of the interface. Therefore the preconditioned Schur complement
method needs less iteration steps than the method without preconditioner. The alternating Schwarz
method reaches the larger error bounds quite fast, but then the convergence deteriorates. Thus the
preconditioned Schur complement method gives the best results for this case. But the convergence
of the alternating Schwarz method can be accelerated by a relaxation procedure (cf. F.-C. OTTO
[Ott99]).
8.2 Numerical results
139
method
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
S
RR
AS
%
7
2
3
10
10
8
12
22
13
14
39
18
18
54
22
16
60
27
%
error
11
4
6
15
11
13
18
22
23
21
39
31
22
58
40
25
62
48
%
–
5
10
–
16
70
–
30
–
–
50
–
–
–
–
–
–
–
%
%
energy
error 7
2
3
9
10
7
11
22
13
12
39
18
13
54
24
15
60
29
12
5
7
16
15
–
20
28
–
–
48
–
–
–
–
–
–
–
Table 8.3: We consider case (a) of Example 6.2.
is decomposed into
squares. We use the
. The
. The
checkerboard pattern (6.29) with mesh size reaction coefficient is given by number of iteration steps is plotted, which is needed to achieve certain error bounds. The case, that the
error bound is not reached within 100 steps, is denoted by ’–’. We use the following notation: ’S’ denotes the
Schur complement method without preconditioner, the preconditioned Schur complement method is denoted
by ’RR’, and the alternating Schwarz method is given by ’AS’.
140
Comparison of some nonoverlapping domain decomposition methods
#
Finally we analyze the three methods in dependence on the number of subdomains. We use the
and
same setting as in Table 8.2. But this time the mesh size is always given by we just test case (a) of Example 6.2. The results can be seen in Table 8.3. As expected, for
all methods the number of needed iteration steps increases for more subdomains. For the Schur
complement method this behavior is covered by the theory. The condition
numbers increase like
(cf. (6.31))
(cf. (6.30)) for the Schur complement equation and like for the preconditioned algorithm, where is decomposed into sub-squares. This explains,
why the number of iteration steps for the preconditioned algorithm becomes larger than without
preconditioner for a larger number of subdomains.
Chapter 9
Summary and Outlook
This thesis deals with the three-field formulation of the advection-diffusion equations. Because of
the possible appearance of layers a straightforward discretization, following the line of F. B REZZI,
D. M ARINI [BM94], is not feasible.
In chapter 2 we started with the single domain problem. We derived a nonstandard method for elliptic problems with inhomogeneous boundary conditions. The boundary conditions were worked
in with the help of Lagrange multipliers. Due to additional stabilization terms the Lagrange multiplier space on the boundary could be chosen almost arbitrarily. Moreover we proved an optimal
a priori estimate. Then, the predicted convergence rate were numerically validated. Finally we
derived an a posteriori estimate for the scheme. So far in the literature these schemes neither have
been extended to the nonsymmetric case, nor have extensive numerical studies been carried out,
nor was an a posteriori estimate shown. In this thesis we closed this gap. But further research and
numerical experiments have to be done in order to verify the sharpness of the a posteriori estimate.
In the next chapter we generalized the scheme of chapter 2 to the multi-domain case on the continuous and the discrete level. The stabilized scheme enables an almost arbitrary choice of the
different function spaces. Using the results of the previous chapter, again, we could derive an optimal a priori estimate. Finally, we could verify the theory with the help of numerical experiments.
Again the generalization to the nonsymmetric case is new. Furthermore, we introduced an abstract
framework for the three-field formulation.
In the next two chapters we proposed two different strategies to split the global problem into a
sequence of local problems, which can be solved in parallel. Thus the three-field formulation
can be seen as a basis for a unified presentation of different domain decomposition methods for
nonoverlapping subdomains.
The first algorithm is based on the Schur complement equation. We could derive an adapted Schur
complement equation for our stabilized scheme. Then, this equation was solved with the help of a
GMRES algorithm. Due to the stabilization terms the results differ slightly from the results, which
are known from the conforming case. In many cases with stabilization we obtained better results
than without stabilization. But the diffusion dominated case showed clearly the necessity of a
preconditioner. Therefore we introduced a preconditioner, which is built up by local Robin problems. Hence, the preconditioner can be applied in parallel, too. Unfortunately, the preconditioned
algorithm deteriorated for a larger number of subdomains. Therefore the introduction of a coarse
space is mandatory. But this is the subject for further research. Further open problems are sharp
estimates of the condition of the stabilized Schur complement equation and its preconditioned
version in dependence on the diffusion coefficient .
142
Summary and Outlook
The second domain decomposition method was an alternating Schwarz method. Using the approach of P. L E TALLEC and T. S ASSI, the works of G. L UBE ET. AL were generalized to the
non-conforming case. Although a convergence proof is missing for the discrete case, the numerical results are very promising. Moreover we compared different choices for the transmission
conditions. We observed, that the choice of G. L UBE ET. AL works best. This result was also
confirmed by the Fourier analysis of chapter 8.
In the last chapter the two presented domain decomposition methods were compared. First, we
analyzed the methods with the help of a Fourier analysis in the case of two subdomains. It could
be shown that the preconditioned Schur complement method possesses much better convergence
properties than the alternating Schwarz method. Unfortunately, this could not be completely validated by the numerical results. This has two reasons. On the one hand a larger number of subdomains requires a coarse space correction in order to ensure a global transport of information.
On the other hand the proposed preconditioner does not take the additional stabilization terms into
account. Thus, one of the most important tasks for the future is a more suitable preconditioner for
the Schur complement equation.
Summarized it can be stated, that we proposed a new multi-domain formulation for the advectiondiffusion equation, which allows to combine different nonmatching meshes. Moreover, it was
shown, that the multi-domain formulation can be effectively decoupled by two different classes of
domain decomposition methods. Furthermore, we showed some ideas, how this technique can be
extended to more complicated equations like the Oseen equations.
Part IV
Appendix
Appendix A
Functional Analysis
In this chapter we summarize the abstract framework. First some basic results about Hilbert and
Banach spaces and their dual spaces are collected. Then the closed range theorem will be introduced and will be used to prove the well-posedness of a class of linear mixed problems. Finally we
extend the results to two-fold saddle point problems. Two-fold saddle point problems consist in
two coupled, linear mixed problems. We derive the well-posedness of these problems and derive
an a-priori estimate for conforming approximations.
A.1
Some basic results
+ " " +
" " " ")'
" #
+
Let be a normed real vector space with norm . The dual of
consists of linear bounded
& . The action of a functional functionals on and will be denoted by
on
an element
is given by ! . The space
is a Banach space and is provided
with the standard norm
/
7
?A
;=
(A.1)
>
The first classical result, which is used, is the Riesz Theorem:
Theorem A.1 Let
exists a unique . Then for each + there
be a real Hilbert space with inner product such that
This defines an operator
+
given by
#
.
is an isometric isomorphism.
Proof: K. YOSIDA [Yos95], ch. III/6.
The second classical result is the Lax-Milgram Lemma. This theorem is very important in the
theory of Finite Element methods.
Theorem A.2 Let
X-elliptic, i.e.
then for each
+
be a real Hilbert space. If a linear operator
'" '" %
there exists a unique with " '" " *% #
*%
%
%
&
"
/
and
+ is coercive or
146
Functional Analysis
Proof: K. YOSIDA [Yos95], ch. III/7.
###
Now, starting from given real Hilbert spaces
space.
H< Lemma A.1
)
,
we construct a new Hilbert
is a Hilbert space with inner product
; H< and induced norm
; )" " H< " "
# # # # # # #
# # # The proof is obtained by simple verification of the axioms. Next we show that the dual of a Hilbert
space is a Hilbert space itself.
+
be a real Hilbert space. Then
Lemma A.2 Let
which is equal to the standard norm (A.1).
Proof: Let
+
is a Hilbert space with an induced norm,
be the Riesz operator of Theorem A.1. Then it is easy to see that
+
+ . Furthermore, using the notation " " + we comdefines a scalar product on
pute
+
" " " " " "
")1" ")'"
" *"
+ . Finally it is easy to verify that + is a Banach space. Therefore the assertion is
for /
/
7
?A
;=
!
7
>
?A
;=
27
>
;=
?A /
/
/
/
>
proved.
In addition, we need a technical lemma about the decomposition of Hilbert spaces, which is important in the theory of linear mixed problems.
Lemma A.3 Let
be a Hilbert space and
M be a closed subspace. Then
is a closed subspace. The decomposition
holds true.
Proof: K. YOSIDA [Yos95], ch. III/1.
A.2 Closed Range Theorem and applications
A.2
147
Closed Range Theorem and applications
We now introduce the Closed Range Theorem and apply it to linear mixed problems. Examples
for linear mixed problems are the Stokes equations or the Oseen equations (cf. chapter 5). But we
apply the theory to weakly enforced boundary conditions (Part I) and to the three-field formulation
(Part II). The theorem is essential for the question, whether linear mixed problems are well-posed
or not.
and be normed spaces. By
First we have to introduce some notations and definitions. Let
we denote the dual space of
and is the duality pairing between
and
. For
&
the dual operator
& is defined by
+
+
+
+
+
0 %
0 %
0 + 0 + #
for the range of 0 and 0 0 In addition, we use the notation 0 0 for the kernel of 0 .
)
" "
" ?"
Theorem A.3 (Closed Range Theorem)
and
be two real Banach spaces with norms and , respectively, and let
Let
&
be a linear, bounded operator. Then the following properties are equivalent:
0 %
0 is closed.
+ is closed.
(ii) The range 0
(iii) 0 + 2 + 0 + ;
+ M + + + 2 0 .
(iv) 0
(i) The range
,
)
)
Proof: K. YOSIDA [Yos95], ch. VII/5.
Now we apply this theorem to linear mixed problems:
+ be a real Hilbert space and a real, reflexive Banach space. Let % + be two linear mappings. Given + and + , we call the
Find +
+
in (A.2)
+
in
Definition A.1 Let
&
and (
problem
$
&
%
$
$
*
%
(
$
(
$
a linear mixed problem.
+ ?
Next we define the polar set of a vector space by . For the main theorem we need the following Lemma. The proof follows V. G IRAULT, P.-A.
R AVIART ([GR86], I.4.1) and D. B RAESS ([Bra92], III.4.2 ).
3
Lemma A.4 The three following
properties are equivalent:
(i) there exists a constant
such that
5
7:9 ; =
>
BC7 - ; =
?A >
)" *" " "
*(
-
9
3
(A.3)
148
Functional Analysis
+ is an isomorphism from onto and
" + " " " onto + and
is an isomorphism from " 2" ")*" #
(
(ii) the operator
$
(
(iii) the operator
3 *)
43
(
*(
9
- /
*)
43
3*(
(A.4)
$
9 /
(
$
-
*)
*(
(A.5)
Remark A.1 The condition (A.3) is very important for mixed problems. It is called BabuškaBrezzi condition.
+ ")*"
Proof: First we prove that the properties (i) and (ii) are equivalent. (A.3) is equivalent to
" + "
3
" " #
")*"
+ is an
Therefore (A.3) is equivalent to (A.4) and (ii) implies (i). Now it remains to prove, that
+ is a bijective operator from onto + isomorphism. Because of (A.4) it is obvious, that
+
+
with a continuous inverse. Therefore is a closed subspace of . Applying the Closed
Range Theorem yields
+ which proves the first part.
Next we show that (ii) implies (iii). For a linear, continuous operator
. Thus
+ by % % . It is obvious, that we define
(ii) implies the existence
+
with . Then we get from (ii)
of an element
%
+
%
" " " " " " " " ")2"
*(
- /
(
?A -
?A *(
9
-
$
(
(
*(
*)
*)
B
*(
3
$
(
B
B
B
*(
*(
-
B
*(
B
$
*)
*(
B
B -E/
(
-
B - /
E7 - ; =
?A
>
+ " "
and therefore
B 3
" B " )" 2"-
#
"
"
. It remains to show that is also surjective. Applying Theorem
Clearly is injective on + is closed.
A.3 we can show analogously to the first step that the range of
+ and with the help of of the Closed Range Theorem we get
Property (ii) implies that + + + #
" *"
(
9 /
*(
B
*(
(
B
7 <
9 ;=
?A >
*)
*(
*)
*(
" "
9
C7
9 / ; =
?A
>
$
*(
*(
$
)
*(
$
Thus ( is an isomorphism.
Finally we show that (iii) implies (i). We get the assertion by
*(
(
(
)
*)
"* "
BC7
D D F F
?A
*(
;=
>
%
" 2"
(
B27 - ; =
?A >
Now we formulate and prove the main theorem of this section:
*(
3
")2"
#
$
A.2 Closed Range Theorem and applications
149
% + satisfies the Babuška-Brezzi condition (A.3) and + ")*" + + the problem (A.2) has a unique solution with
then for each " 1" " 1" % 0 " " " " " 2" #
(A.6)
+
+
The mapping is an isomorphism from onto .
Proof: It follows from the Babuška-Brezzi condition (A.3) and Lemma A.4 that there exists a
unique solution of with
" " % " *" #
Theorem A.4 If
fulfills
(
&
+$
+
*%
+
$
-
*
%
*)
*(
(
)
$
$
9 /
-
Then we consider the following auxiliary problem
$
3
Find
9 /
- /
*
&
*(
3
9
)
-
%
*(
*%
*%
$#
)
*(
Because of the Lax-Milgram Lemma (cf. Theorem A.2) the auxiliary problem has a unique solution ) *( with
+
Thus " " %
L " " M" 4" " " U #
+
in + in
-
is a solution of
-E/
%
(
*)
" '" % 0 %
*(
$
+
3
" " " " "*" #
(A.7)
Applying Lemma A.4 again yields a unique solution of
+ because . Furthermore, we get by virtue of (A.7) and Lemma A.4
" 1" % " '" % K" " M" 4" " '" % 0 " " " " " *" #
which fulfills the a priori estimate (A.6).
Hence, problem (A.2) has a solution Finally we show that only the trivial solution solves the homogeneous problem of (A.2) with
. Adding the two equations of (A.2) with test functions and yields
which gives because of . Moreover, we get
with the a priori estimate
-
%
(
%
*)
3
9
%
*(
9 /
- /
$
%
+
3
- /
-E/
%
*
-
3
%
-E/
9 /
$
*%
)
and therefore unique solution.
!
*(
*(
* with the help of condition (A.3). So we have shown that (A.2) possesses a
Unfortunately we need more general linear mixed problems. We call the following problem a
two-fold saddle point problem because it consists of two linear mixed problems.
150
Functional Analysis
Banach space. Furthermore,
+
be real Hilbert+ spaces and a reflexive
+
, and 0 are linear, bounded operators. For
such that
+ + , and + find +
+ + in +
(A.8)
0
in
+
#
0 in
Definition A.2 Let
&
suppose %
given
,
$
(
&
$
$
& $
%
(
$
(
$
Now we have
Theorem A.5 Let (
and
0
43
satisfy the inf-sup conditions
3
(A.9)
")2" " " and
" "0 " " #
(A.10)
0 , i.e. there exists
Furthermore, let be coercive on such that
N ! '")2" #
Then there exists a unique solution of (A.8) with
(A.11)
" 1" " " " " - " " "K " " " #
5
7:9 ; =
+
+
9
>
7 <
9 ;=
?A >
%
*(
*%
)
$
9
-
-
9
-
5
;=
7
*(
BC7 - ; =
>
?A >
9 /
-E/
/
The proof is based on O. S TEINBACH [Ste00].
Proof: (i) Because of the inf-sup condition (A.10) and Lemma A.4 there exists a unique
*)
and
with
0 0
" " - " " #
+ for all 0 we consider the following auxiliary problem: Find
(ii) Because of 0
0 such that
+
+
in +
+ #
(A.12)
in 0
DE
/
)
)
%
(
(
(
*)
(iii) Next considering the modified inf-sup condition
")*" :" "
0 of (A.12) with
we can apply Theorem A.4. This yields a unique solution " 1"" " - "K "" + " - "K "M" " " " #
we get by virtue of Lemma A.4 a unique with 0 + (iv) Since 0
and
" " - "K 1" - "K " " " " " #
7
D 5 F ;=
BC7 - ; =
>
?A >
)" *" " "
3
*(
-
5
7:9<;=
9
*(
BC7 - ; =
>
?A >
-
(
(
9
)
*)
(
(
A.2 Closed Range Theorem and applications
Defining
151
we get a solution of (A.8) satisfying (A.11) where we used
" - " R" " " #
(v) Now we show, that the solution is unique by proving that only the trivial solution solves (A.8)
be uniquely
with
, and . Let be a solution of (A.8).
. Becausecanof (A.10)
0
0
decomposed into with
and
there
holds
(cf. Lemma A.4). Then the first two lines of (A.8) imply
+
+
in in 0 + #
+ implies . Therefore the
. And finally 0
Now Theorem A.4 yields and
" " "
*)
)
%
(
(
*)
solution is unique.
$ and
Next we consider a conforming finite dimensional approximation. Let ,$ be finite dimensional subspaces. Then the discrete two-fold saddle point problem is
given by: Find $ such that
+ 0+ 0 % (
(
for
+ , +
+
and
5 7
0
Proof:
in
5 5 ;=
*(
>
>
B
5
0
In addition, we require that % is coercive on
where is the discrete analogue of .
Then there exists a unique solution +
(A.13)
43
5
7 9 C;=
in in $ and
Theorem A.6 Let (
where
++
+.
and 0 satisfy the inf-sup conditions
5 5 5 5 ") " 4E " " $
"
"
-
"
") 1
"-
5
B 5 7 - 5
$
7 - 2;=
?A
>
9
8
(A.14)
9
-
+
#
0
8
" " 9 " G" 7 9 5C;=
?A
>
(A.15)
*( $ ,5
7 9
9
8
0 )
of (A.13) with the error estimate
" " 5 5 "
5 5 " " "
3
,5
6 :7
"
(A.16)
is the solution of (A.8).
Completely analogously to Theorem A.5 the existence and uniqueness of the solution
$ of (A.13) can be proved. The error estimate (A.16) can be found in
O. S TEINBACH [Ste00], Theorem 1.6.
Appendix B
Function spaces
In order to develop the theory of the three-field method a thorough treatment of various Sobolev
spaces is necessary. Therefore the definition and properties of some important function spaces are
recalled. This section is a supplement and a glossary of terminology and results used in the text.
The presentation is based on the books of R.A. A DAMS [Ada75], P. G RISVARD [Gri85] and W.
M C L EAN [Lea00].
B.1 Smooth functions
All functions are defined on subspaces of
equipped with the usual inner product
, , and are real valued. The vector space
;
### ### H< , .
and the related norm
To keep the presentation short we use the multi-index notation, i.e. for each vector
# # # + we define
( (F
T ;H< D
& Q $& .
for sufficiently smooth functions Now we can define the following vector spaces of continuous functions.
be an open subdomain and .
Definition B.1 Let
+
+
+
+
+
+
The set of continuous functions on
is given by
0 is continuous
The set of -times continuously differentiable functions on
0 M +
#
is defined
by
+
0 2 #% #
is
154
Function spaces
0 0 , which together with its derivatives
of .
Let 0
be the vector space consisting of functions, which are infinitely
often differen tiable and possess a compact support in . We also use the notation 0 .
Note that 0
, + , is a Banach space with the norm
" R" D F . (cf. R.A. A [Ada75] , ch. 1.26).
for a bounded open set
Let
be the set of functions belonging to
can be extended continuously to the boundary
?A
7
DAMS
Next Hölder spaces are defined, which we need to describe the smoothness of boundaries.
% %
0 Definition B.2 For
with
and consists of functions in
#
(B.1)
the Hölder space
0
" " D F " " D F ; < Again it can be shown that 0
with the norm (B.1) is a Banach space (cf. R.A. A
[Ada75], ch. 1.27).
can be described:
Now the smoothness of the boundary 4% % and , . We say that its boundary
Definition B.3 Let be an open subset of ,
is of class 0 , if the following conditions are satisfied:
there exist a neighborhood of in
and new orthogonal coordinates
For every # # # such that is a hypercube in the new coordinates:
# # # % % .
and there exists a function 0
with
. M # # # % % ?
A DAMS
and such that
0
A boundary of class
Remark B.1
. % . # # # . .. .. #
is called Lipschitz boundary.
0
and a bounded open domain
For
the boundary
is of class if and
only if is a -submanifold with boundary in .
Is the boundary
a Lipschitz boundary, then the closure is a Lipschitz submanifold with
boundary in . The converse statement is not true (cf. P. G RISVARD [Gri85], Th. 1.2.1.5).
0
A domain with Lipschitz boundary satisfies the uniform cone property and vice versa (cf. P.
G RISVARD [Gri85], Th. 1.2.2.2).
B.2 Lebesgue spaces
155
B.2 Lebesgue spaces
with
Now we define Lebesgue spaces and collect some properties of these spaces. We consider
the Lebesgue-measure . If
is a measurable set, two measurable functions are called equivalent, if a.e. (almost everywhere) in . An element of a Lebesgue space is
an equivalence class.
is defined as the set of
Definition B.4 Let
be a measurable set and .
such that " " A D F with
equivalence classes of measurable functions Q
(B.2)
" " A D F %! % ifif #
% % . Then with the norm (B.2) is a Banach space.
Theorem B.1 Let be % %
7
?A
Proof: R.A. A DAMS [Ada75], ch. 2.10.
Note, that is a Hilbert space with the inner product
A D F $#
Sometimes we simply write or . The corresponding norm is denoted by " " or " #" .
B.3 Distributions, weak derivatives and Sobolev spaces
The main focus of this section is the definition of Sobolev spaces. To this end we need the concepts
of distributions and weak derivatives. First we consider the space of distributions. Let be a
measurable domain in .
belonging to
converges in the sense of
We say that a sequence of functions
the space
to the function
, if there exists a compact set
in such that
supp
for all and
<
0 0 +
uniformly on
from above.
for each multi-index . We provide the space
+ Definition B.5 The dual space
of
If
and
, the value of
+ with the induced topology
is the space of distributions.
$#
we define a distribution by
! $#
at
will be denoted by
For a function
This yields an injective embedding of
into
Now the concept of derivatives can be generalized:
+ and we identify with .
156
Function spaces
Definition B.6 For
derivative, if
+ +
and a multi-index
! + we call
$#
! the distributional
In the next step Sobolev spaces are introduced:
, Definition B.7 The Sobolev space
Here
, is defined by
+
2 % #
is understood as the distributional derivative.
Theorem B.2
with scalar product
( ; and corresponding norm
" 1" ( ( $#
is a Hilbert space.
Proof: R.A. A DAMS [Ada75], ch. 3.2.
In addition, sometimes we need the seminorm
Q
;
( < $#
, is given by the completion of w.r.t. the norm "
Definition B.8 , Clearly , , is a Hilbert space
to the scalar product with respect
,
+
is often denoted by space of .
(
( we define by
Definition B.9 For M % #
( , , is a Banach space with the norm (
Theorem B.3 " '" ( % % $#
" ( .
( . The dual
+
Proof: R.A. A DAMS [Ada75], ch. 3.2.
?A
7
Sobolev spaces have some interesting properties. For us the Sobolev Imbedding Theorem is very
important:
B.4 Trace theorems and Sobolev spaces of fractional order
157
Theorem B.4 Let
be a bounded domain with Lipschitz boundary and suppose
Then the following continuous imbeddings are true:
For For For we obtain we obtain there holds
for all .
for all % .
0 .
.
Proof: R.A. A DAMS [Ada75], ch. 5.4.
B.4 Trace theorems and Sobolev spaces of fractional order
be a measurable set with Lipschitz boundary
. The boundary
In this section let
will be denoted by .
On the -dimensional set it is also possible to define Sobolev spaces:
Q is defined by
Q Q ( (
where the seminorm Q is given by
Q ( % % Q $#
Q Theorem B.5 with the scalar product
(
%
Q % % of
Definition B.10
(B.3)
(B.4)
is a Hilbert space.
Proof: P. G RISVARD [Gri85], ch. 1.3.3.
-dimensional relative open subset. Then we define
Q Q with Q with norm
" 1" O Q D Q F Q " " O Q D F Q $$#
Q
Q Q the zero extension of into
Now we construct a particular subspace of . For will be denoted by . So we can define:
Let
be a proper, connected
5
158
Function spaces
Definition B.11
Q is given by
Q $ $ Q #
Q By a direct calculation (see P. G RISVARD [Gri85] or J. X U , J. Z OU [XZ98], ch. 4.1) for
we obtain the existence of constants , such that
0
@" 2 "#Q ( % )" 2" O Q D Q F % 0 " 2 "#Q ( (B.5)
with
")*" O Q D Q F ")*" Q ( Q Q % (B.6)
is a positive function which behaves like the distance between and . Notice
where that
(B.7)
O Q D Q F Q ( Q Q % Q Q $ . Therefore Q is a Hilbert space (cf. the norm equivadefines a scalar product in Q
lence
(B.5)). The space can also be obtained by Hilbert scaling between the spaces and
(cf. J. B , J. L
[BL76]).
Q
Q
$
Q
+
! and $ Q $ + .
The dual of these spaces are denoted by and 0 . Then we can define the
Next we present some trace theorems. Let be 0 traces
& ERGH
ÖFSTR ÖM
where is the normed outward normal of . These trace operators can be extended:
0 0
Theorem B.6 Let
be an
. Then the
open, bounded domain with boundary
trace mapping defined on extends uniquely to a bounded, surjective linear map
Analogously, if the boundary is of class 0
a bounded, linear map
Q $#
to
, the normal derivative extends from 0
Q $#
Proof: P. G RISVARD [Gri85], Th. 1.5.1.2.
Sometimes the simpler notation operator we can characterize the space
is the kernel of Theorem B.7
(B.8)
is used for functions . With the trace
:
, i.e.
$#
)
Proof: P. G RISVARD [Gri85], Cor. 1.5.1.6.
In the context of stabilized norms, we use the following extension of the trace theorem:
B.5 Some fundamental equalities and inequalities
Theorem B.8 Let
and for all 159
Q
.
Q be a bounded, open domain with Lipschitz boundary. Then there holds
%
(B.9)
Proof: P. G RISVARD [Gri85], Th. 1.5.1.10.
Moreover the right inverse of the trace operator exists:
Q Q $#
be a bounded, open domain
with Lipschitz boundary. Then there exists
Theorem B.9 Let
a linear, bounded operator
, such that
Proof: P. G RISVARD [Gri85], Th. 1.5.1.2.
Note that the preceding theorems allow the definition of the following equivalent norm on
" '" Q ( ")2" O Q D F #
Q
5
Q :
(B.10)
B.5 Some fundamental equalities and inequalities
Green’s formula
The first equality is the Green’s formula. We use a variant for weak functions:
Theorem B.10 Let be a bounded open subset of
we have
where &
with a Lipschitz boundary. Then for all
& % # # #
is the -th component of the outward normal vector .
Proof: J. N EC ÁS [Nec67], Th. 3.1.1.
Sometimes a second version of the Green’s formula is useful.
Theorem
be open and bounded with Lipschitz boundary and
B.11 Let
. Then we obtain
for any .
%
with
(B.11)
Proof: C. S CHWAB [Sch98], p. 352, V. G IRAULT, P.-A. R AVIART [GR86], ch. I, Cor. 2.8.
Hence we can use (B.11) to define
B
as an element of the dual space
Q ,+ .
160
Function spaces
Inequality of Poincaré
Now we cite a variant
of the inequality of Poincaré. It allows to estimate the function values of
functions by the first derivatives of functions .
Theorem B.12 Let
, , be a bounded domain with Lipschitz boundary. Further
% . Then the
more let
be a connected part of the boundary of with inequality
" '" ( % 0 $ ( Q . The constant 0 $ depends only on
is true for all with and is bounded by the diameter of .
and
Proof: cf. A. Q UARTERONI, A. VALLI [QV94], Th. 1.3.3.
B.6 Finite Element spaces
In this section we introduce the concept of Finite Elements. We restrict ourselves to the construction of simplicial Finite Elements. A detailed introduction of more general concepts can be
found for example in the books of S.C. B RENNER , L.R. S COTT [BS94] and A. Q UARTERONI ,
A. VALLI [QV94].
Definition
+
,
, Let us start with a bounded, polyhedral domain
a family of partitions of into -simplices, i.e.
. Then we denote by #
7 35
To ensure the continuity of the discrete spaces, defined with the help of the partitions, we need the
following additional condition:
4% % Definition B.12 A partition of
or share a complete -face,
is called admissible, if two elements
.
and are either disjoint
Remark B.2 The condition of admissibility means, that there are no hanging nodes in .
and as the diameter of the largest ball
Denoting as the diameter of a -simplex
inscribed into , we can formulate another important property of the partition :
Definition B.13 The partition is called shape regular if
? A
?7A
35
- #
B.6 Finite Element spaces
161
Furthermore, is called quasi-uniform, if
where is given by ?A
-
743 5 -
.
Remark B.3 The first condition ensures that asymptotically the simplices do not degenerate. The
meaning of quasi-uniformity is, that the size of all simplices of one partition is asymptotically
equal up to a constant not depending on the parameter .
M +
Now we can define the Finite Element spaces:
where is the set of polynomials of degree at most . Sometimes we need Finite Element spaces
which vanish on a part of the boundary :
( #
Properties
( .
Here we report about some important properties of the spaces resp. paragraph we assume that all partitions of are admissible and shape regular. For
. Therefore we can restrict ourselves to the second space.
obtain ( The following inverse inequality is very useful.
( % (
Lemma B.1 There exists a positive constant
such that
)
, for B (
B
independent of
D F 4 ( ( ( ,
and #
In this
we
(B.12)
.
Proof: S.C. B RENNER , L.R. S COTT [BS94], Lemma 4.5.3.
Furthermore, we need the following local variant of the trace theorem:
Lemma B.2 The inequality
holds for
+
, Q
(
)" 2" ")2" ( ")2" Q ( " *" Q ( and .
Proof: R. V ERF ÜRTH [Ver98], Lemma 3.1.
The next issue are the approximation properties in Sobolev spaces of our discrete space Since functions in
( .
are not continuous in general, we cannot define the usual interpolation operator. Therefore we
consider the so called quasi-interpolation operator introduced by P. C LEMENT [Cle75]. A similar
operator was introduced by L.R. S COTT, S. Z HANG [SZ90]:
162
Function spaces
( with
( - ( !#" ( - ' Q ( ! ( " *" ( - ")*" ( ! " for +
, % *$+ , % $ . - and - are defined by
. - - - " "
( "
"
" Lemma B.3 There exists a linear, bounded operator
and
is a face resp. edge of
(B.13)
(B.14)
(B.15)
.
Proof: The estimates (B.13) and (B.15) are standard. Using Lemma B.2 the inequality (B.14) is
obtained by
;
(
" " < - ; < " " " " " " - ( ( ( - ( !#"
where we also have used the Cauchy Schwarz and the interpolation estimate (B.13).
, are continuous, we can replace the quasiCorollary B.1 If the functions
, %
interpolation operator by the standard nodal interpolation operator. Then the patches and - can be reduced to resp. .
Due to the Sobolev Imbedding Theorem (cf. Theorem B.4) the functions of
are continuous,
%
if is larger than , where is the space dimension.
The results of the quasi-interpolation operator given in Lemma B.3 can be simply extended to the
stabilized norm
with constants proved:
" ( " K 2" "
, a domain and a function , a.e.. Then it can be
2" - ") Q " !#"
)" 2" ( - IQ ! " " " ! (
" " ( - " " !#"
" "
. is defined by
for - , ,
,
Q Q
and or is a face of + #
Lemma B.4 The quasi-interpolation operator of Lemma B.3 satisfies
")
B.6 Finite Element spaces
163
Proof: The first three inequalities are proved by R. V ERF ÜRTH ([Ver98], Lemma 3.2). The last
inequality is a simple result of the inverse inequality (B.12)
and
( - " 2" ( - Q " " !#"
( % Q " " - Q " " !" #
Combining these estimates proves the last inequality.
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Notation
S YMBOL
D EFINITION
PAGE
Part I: The single domain case
-5.
35.
75 .
7
diffusion coefficient
7
flow field
7
reaction coefficient
7
bilinear form
8
0 such that % 0 .
0 such that 0 .
4 -. and 43.
extension operator
on M inflow of
8
linear form
differential operator
bounded Lipschitz domain
8
8
8
8
9
9
10
11
" " Q +
L U
" " ( " " admissible decomposition of
14
discrete finite element space
14
discrete space of the Lagrange multipliers
14
quasi-interpolation operator
14
11
12
13
13
to be continued
174
Notation
S YMBOL
-
patch of
-
+
7 35 -
PAGE
14
14
15
15
bilinear form of the SUPG method
16
linear form of the SUPG method
16
stabilization
parameter
16
16
bilinear form of the stabilized scheme
19
" " " " 435 " " ( " )" " )" ( ! % 35 " B" ( " " ( 3 5 E " " " " " " " " , stabilization parameters
Q Q or is a face of a +
19
set of all faces resp. edges of the decomposition jump on a face 26
mesh size of 31
mesh size of the Lagrange multiplier mesh
31
linear form of the stabilized scheme
3
7
7
7
D
+
3
resp. (
1 restriction of to the boundary
0
+
Peclet number
3
-
" " " #" " #"
(
" #" D EFINITION
D
19
19
19
19
24
25
26
Part II: The three-field formulation
Q
+ "#" (( +
Q +
+ 0 H < + Q subdomains of
Q if ; else + Q Q Q norm of + Q inner product of + Q +
dual
space
of
H< 42
42
42
42
42
42
42
43
43
to be continued
175
S YMBOL
D EFINITION
+ Q M there exists norm of 43
norm of
43
extension operators
43
,
0
,
Q Q dual product on + + inner product with corresponding
norm of
45
45
operators corresponding to the three-field formulation
discrete approximations of
triangulations of , and
46
51
53
maximum mesh sizes
% %
53
Lagrange multiplier space on
)
% %
54
associated with
54
such that
5
set of all edges given by the finite element meshes restricted to
space of Lagrange multipliers for the mortar method
, " " ( " " (( " "
5 54
nonmortar side of
43
43
( 43
norm of
%
on H< " "
" "
" "
" "
PAGE
)
54
54
global quasi-interpolation operator
+ streamline diffusion
norm
in
54
57
3 ! % + Q 58
bilinear form corresponding to stabilized local Dirichlet problems
60
7
3
linear
form corresponding to stabilized local Dirichlet problems
"
" @ ( " " ( 7
3
3 E" " 3 " " " " ( ! 7
+
59
59
60
61
% "
" ( bilinear form corresponding to the stabilized three-field formula-
61
61
61
tion
to be continued
176
Notation
S YMBOL
D EFINITION
" " H< L " " ( " " ( U
H
<
3 ! %
" "
.J 7
" " ," " ," " 62
viscosity
79
bilinear form corresponding to the Oseen equations
79
, ,
+
,
bilinear form
corresponding
to the Oseen equations
$
PAGE
3
$
79
80
function spaces for the three-field formulation (5.8)
, and
H< ###
80
norms for
M
81
=< ! 81
81
Part III: Nonoverlapping Domain Decomposition Methods
operator
# " "
( +
+
the
local
+
(cf. Lemma 3.7)
corresponding
Dirichlet
problems
88
88
88
Steklov-Poincaré operator
88
right hand side of the Schur complement equation
88
local Steklov-Poincaré operator
90
Robin–Robin preconditioner
92
prolongation resp. restriction operators
92
solution of local problems
98
discrete Steklov-Poincaré operator
98
right hand side of the discrete Schur complement equation
98
discrete norm for 2
99
discrete local Steklov-Poincaré operator
local
S( Krylov space
102
-projection
+
100
error indicator
102
.
111
113
120
to be continued
177
S YMBOL
contraction
rate of the one-dimensional Dirichlet-Robin alg.
4
125
126
128
contraction rates for the two-dimensional case for the precondi-
(
PAGE
contraction rate for the one-dimensional case
D EFINITION
( $ 129
tioned Schur complement equation
*)
contraction rates for the two-dimensional Robin-Robin algorithm
130
134
Part IV: Appendix
0 0
)
0 0 0 (
0 ( 0 " "BA D F " "B " "
+ ( ( (
" Q " Q , Q
Q , Q ( " " O Q D F
( range of an operator 0
kernel of an operator 0
polar set QQ TT
146
147
147
147
153
set of -times continuously differentiable functions
154
space, consisting of infinitely differentiable functions with com-
154
pact support in
Hölder space
154
smoothness of the boundary
154
norm of Lebesgue space
defined in the domain
inner product resp. norm of
155
155
155
space of distributions
155
Sobolev spaces
156
norm resp. seminorm of
156
trace spaces
157
Q inner product resp. norm of 158
Finite Element spaces in
161
dual spaces of the trace spaces
of degree
158
Index
a posteriori error estimation, 25
advection-diffusion-reaction problem, 7
alternating Schwarz method, 1
anisotropic mesh, 38
ARN-method, 115
Green’s formula, 159
Babuška-Brezzi condition, 12, 47, 148
Black-Scholes model, 7
BPS-preconditioner, 101
bubble function, 15
inequality of Poincaré, 160
inflow of the boundary, 10
inverse inequality, 14, 57, 161
iteration-by-subdomain algorithm, 4
iterative substructuring methods, 2
CG method, 2
Closed Range Theorem, 147
coercive, 145
computional fluid dynamics (CFD), 7
convection dominated case, 15
crosspoint, 55
degrees of freedom, 68
Delaunay algorithm, 68
Dirichlet-Neumann method, 2
discontinuous Galerkin method, 11
distribution, 155
distributional derivative, 156
domain decomposition method
nonoverlapping domain
decomposition method, 1
overlapping domain
decomposition method, 1
dual operator, 147
elliptic extension, 9
energy norm, 13
extension operator, 43
Hölder space, 154
hanging nodes, 160
hyperbolic limit, 17
Jacobi method, 102
kernel, 147
Krylov method, 2
Krylov subspace, 102
Lax-Milgram Lemma, 145
layer, 10
Lebesgue spaces, 155
linear mixed problem, 12, 13, 147
Lipschitz boundary, 154
mesh
admissible mesh, 160
quasi-uniform mesh, 161
shape regular mesh, 160
mortar, 41
mortar method, 54
mortar side, 54
multi-index, 153
fictious domain approach, 3
Finite Element space, 161
Navier-Stokes equations, 7, 79
Neumann-Neumann preconditioner, 2
Nitsche’s method, 11
nonmortar side, 54
geometrical conform decomposition, 54
GMRES method, 2, 69
oscillations in crosswind direction, 16
Oseen equations, 79
180
INDEX
outflow, 59
wavelet discretization, 11
Peclet number, 15
polar set, 147
pressure, 79
X-elliptic, 145
quasi-interpolation operator, 57, 161
range, 147
reduced problem, 9
Richardson iteration, 94
Riesz Theorem, 145
Robin-Dirichlet method, 2
Robin-Robin method, 2
Robin-Robin preconditioner, 2, 93
saddle point problem, 12
Schur complement equation, 2, 87
Schwarz method, 1
additive Schwarz method, 1
alternating Schwarz algorithm, 111
multiplicative Schwarz method, 1
singularly perturbed case, 9, 42
slave side, 54
Sobolev Imbedding Theorem, 156
Sobolev spaces, 156
SOR algorithm, 102
SSOR algorithm, 102
Steklov-Poincaré operator, 2, 87
discrete Steklov-Poincaré operator, 98
local Steklov-Poincaré operator, 90
streamline diffusion method, 16
strip-wise partition, 111
subgrid scale models, 16
SUPG method, 16
three-field formulation, 3, 41, 45
for the Oseen equations, 81
stabilized three-field formulation, 58
trace inequality, 158
trace space, 157
two-fold saddle point problem, 145, 149
uniform cone property, 154
velocity field, 79
viscosity, 79
Curriculum vitae – Lebenslauf
Persönliche Daten:
Name:
Gerd Rapin
Geburtsdatum:
2.6.1973
Geburtsort:
Haselünne
Familienstand:
ledig
Eltern:
Margret Rapin, geb. Sandhaus, Lehrerin
Gerhart Rapin, Ingenieur
Schulbildung:
8/1979 - 4/1982
Paulus Schule, Haselünne
4/1982 - 7/1983
Grundschule Monheim
8/1983 - 12/1986
Gymnasium Donauwörth
1/1987 - 6/1992
Gymnasium Nordhorn
6/1992
Abitur
Studium:
10/1993 - 10/1995
Grundstudium der Mathematik mit Nebenfach Volkswirtschaftslehre an der Georg-August-Universität Göttingen
10/1995
Vordiplomprüfungen Mathematik
10/1994 - 10/1999
zusätzlich Studium der Volkswirtschaftslehre an der GeorgAugust-Universität Göttingen
2/1997
Vordiplomprüfungen Volkswirtschaftslehre
10/1995 - 10/1999
Hauptstudium Mathematik an der Georg-August-Universität
Göttingen
1999
Diplomarbeit: ”Die Navier-Stokes Gleichungen - Zur Problematik von a-posteriori Fehlerschätzern unter Beachtung hydrodynamischer Stabilität”
10/1999
Diplom in Mathematik
Berufliche Tätigkeit:
8/1992 - 10/1993
Zivildienst
seit 11/1999
wissenschaftlicher Mitarbeiter und Doktorand am Institut für
Numerische und Angewandte Mathematik der Georg-AugustUniversität Göttingen
seit 3/2000
assoziiertes
Mitglied
im
Graduiertenkolleg
mungsinstabilitäten und Turbulenz”
3/2002 - 6/2002
Forschungsaufenthalt am Politecnico di Torino (Italien) bei Herrn
Prof. Dr. C. Canuto
“Strö-