Numerical Approximation Methods for Elliptic Boundary Value Problems

396 Pages·2008·2.17 MB·English

Numerical Approximation Methods
for Elliptic Boundary Value Problems
Olaf Steinbach
Numerical Approximation
Methods for Elliptic Boundary
Value Problems
Finite and Boundary Elements
OlafSteinbach
InstituteofComputationalMathematics
GrazUniversityofTechnology
Austria
Originally published in the German language by B.G. Teubner Verlag as “Olaf Steinbach:
NumerischeNäherungsverfahrenfürelliptischeRandwertprobleme.1.Auflage(1sted.)”.
(cid:1)c B.G.TeubnerVerlag|GWVFachverlageGmbH,Wiesbaden2003
EnglishversionpublishedbySpringerScience+BusinessMedia,LLC
ISBN978-0-387-31312-2 e-ISBN978-0-387-68805-3
LibraryofCongressControlNumber:2007936614
MathematicsSubjectClassification(2000):65N30,65N38
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY
10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection
withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper.
9 8 7 6 5 4 3 2 1
springer.com
Preface
Finiteandboundaryelementmethodsbelongtothemostusednumericaldis-
cretization methods for the approximate solution of elliptic boundary value
problems. Finite element methods (FEM) are based on a variational formu-
lation of the partial differential equation to be solved. The definition of a
conforming finite dimensional trial space requires an appropriate decomposi-
tionofthecomputationaldomainintofiniteelements.Theadvantageofusing
finite element methods is their almost universal applicability, e.g. when con-
sideringnonlinearpartialdifferentialequations.Contrary,theuseofboundary
elementmethods(BEM)requirestheexplicitknowledgeofafundamentalso-
lution,whichallowsthetransformationofthepartialdifferentialequationtoa
boundary integral equation to be solved. The approximate solution then only
requires a decomposition of the boundary into boundary elements. Bound-
ary element methods are often used to solve partial differential equations
with(piecewise)constantcoefficients,andtofindsolutionsofboundaryvalue
problems in exterior unbounded domains. In addition, direct boundary el-
ement methods provide a direct computation of the complete Cauchy data
which are the real target functions in many applications. In finite element
methods, the Cauchy data can be computed by using Lagrange multipliers
and by solving related saddle point problems. By combining both discretiza-
tion methods it is possible to profit from the advantages of both methods.
Although the aim of this book is to give a unified introduction into fi-
nite and boundary element methods, the main focus of the presentation is on
the numerical analysis of boundary integral equation methods. Therefore, we
only consider some linear model problems such as the potential equation, the
system of linear elasticity, the Stokes system, and the Helmholtz equation.
When considering the above mentioned elliptic boundary value problems it is
possible to describe and to analyze finite and boundary element methods in
a unified manner. After the description of the model problems, we introduce
thefunctionspaceswhichareneededlater.Thenwediscussvariationalmeth-
odsforthesolutionofoperatorequationswithandwithoutsideconditions.In
particular,thisalsoincludestheformulationofsaddlepointproblemsbyusing
VI Preface
Lagrangemultipliers.Thevariationalformulationofboundaryvalueproblems
is the basis of finite element methods, but on the other hand, domain varia-
tionalmethodsarealsoneededintheanalysisofboundaryintegraloperators.
After the computation of fundamental solutions we define certain boundary
integral operators, and analyze their mapping properties such as bounded-
ness and ellipticity. For the solution of different boundary value problems we
then describe and analyze different boundary integral equations to find the
complete Cauchy data. Numerical discretization methods are first formulated
and investigated in a more abstract setting. Afterwards, appropriate finite
dimensional trial spaces are constructed, and corresponding approximation
properties are given. For the solution of mixed boundary value problems we
then discuss different finite and boundary element methods. In particular, we
investigate the properties of the associated linear systems of algebraic equa-
tions. For this, we also describe appropriate preconditioned iterative solution
strategies where the proposed preconditioning techniques involves both the
preconditioningwithintegraloperatorsoftheoppositeorder,andahierarchi-
cal multilevel preconditioner. Since the Galerkin discretization of boundary
integral operators leads to dense stiffness matrices, fast boundary element
methods are used to obtain an almost optimal complexity of storage and of
matrix by vector multiplications. Finally, we describe domain decomposition
methodstohandlepartialdifferentialequationswithjumpingcoefficients,and
to couple and parallelize different discretization techniques such as finite and
boundary element methods.
Boundary element methods are a well established numerical method for
elliptic boundary value problems as discussed in this textbook. For the sake
of simplicity in the presentation, we only consider the case of linear and self–
adjoint partial differential equations. For more general partial differential op-
erators one has to consider the fundamental solution of the formally adjoint
operator. While the existence of fundamental solutions can be ensured for a
large class of partial differential operators their explicit knowledge is manda-
tory for a numerical realization. For nonlinear partial differential equations
there also exist different approaches to formulate boundary integral equation
methods which are often based on the use of volume potentials to cover the
nonlinear terms.
While there exists a rather huge number of textbooks on finite element
methods, e.g. [5, 21, 31, 41, 57] just to mention a few of them, much less
is available on boundary integral and boundary element methods. For the
analysis of boundary integral operators related to elliptic partial differential
equations we refer to [3, 60, 81, 88, 102, 103] while for the numerical analysis
ofboundaryelementmethodswemention[39,124,126,135].Inaddition,the
references[9,15,19,30,61,77]areonpracticalaspectsoftheuseofboundary
element methods in engineering. In [74, 98, 125, 158] one may find recent
results on the use of advanced boundary element algorithms. For a detailed
description of fast boundary element methods starting from the basic ideas
Preface VII
and proceeding to their practical realization see [117] where also numerous
examples are given.
Since the aim of this textbook is to give a unified introduction into finite
and boundary element methods, not all topics of interest can be discussed.
Forafurtherreadingwerefer,e.g.,to[51,130]forhpfiniteelementmethods,
to [2, 8, 10, 153] for a posteriori error estimators and adaptive finite element
methods, and to [24, 69] for multigrid methods. In the case of boundary el-
ement methods we refer, e.g., to [17, 99, 100, 131, 146] for hp methods, to
[37, 38, 56, 128, 129] for a posteriori error estimators and adaptive methods,
and to [96, 111, 147] for multigrid and multilevel methods. Within this text-
book we also do not discuss the matter of numerical integration, for this we
refer to [55, 67, 87, 93, 115, 123, 132, 148] and the references given therein.
This textbook is based on lectures on the numerical solution of elliptic
partial differential equations which I taught at the University of Stuttgart, at
the Technical University of Chemnitz, at the Johannes Kepler University of
Linz,andatGrazUniversityofTechnology.Chapters1–4,8,9,11,and13can
be used for an introductory lecture on finite element methods, while chapters
1–8,10,12,and13areonthebasicsoftheboundaryelementmethod.Chapter
14 gives an overview on fast boundary element methods. Besides the use as
a complementary textbook it is also recommended for self–study for students
and researchers, both in applied mathematics, in scientific computing, and in
computational engineering.
It is my great pleasure to thank W. L. Wendland for his encouragement
and support over the years. Many results of our joint work influenced this
book.SpecialthanksgotoJ.BreuerandG.Of,whoreadtheoriginalGerman
manuscriptandmadevaluablecommentsandcorrections.Thistextbookwas
originally published in a German edition [140]. Once again I would like to
thank J. Weiss and B. G. Teubner for the fruitful cooperation.
WhenpreparingtheEnglishtranslationIgotmanyresponses,suggestions
and hints on the German edition. I would like to thank all who helped to
improve the book. In particular I thank G. Of, S. Engleder and D. Copeland
who read the English manuscript. Finally I thank Springer New York for the
cooperation and the patience when preparing this book.
Graz, August 2007 Olaf Steinbach
Contents
1 Boundary Value Problems ................................. 1
1.1 Potential Equation ...................................... 1
1.2 Linear Elasticity ........................................ 5
1.2.1 Plane Elasticity ................................... 9
1.2.2 Incompressible Elasticity ........................... 12
1.3 Stokes System .......................................... 12
1.4 Helmholtz Equation ..................................... 15
1.5 Exercises ............................................... 17
2 Function Spaces ........................................... 19
2.1 The Spaces Ck(Ω), Ck,κ(Ω) and L (Ω) .................... 19
p
2.2 Generalized Derivatives and Sobolev Spaces................. 22
2.3 Properties of Sobolev Spaces.............................. 25
2.4 Distributions and Sobolev Spaces.......................... 29
2.5 Sobolev Spaces on Manifolds.............................. 35
2.6 Exercises ............................................... 39
3 Variational Methods....................................... 41
3.1 Operator Equations...................................... 41
3.2 Elliptic Operators ....................................... 46
3.3 Operators and Stability Conditions ........................ 48
3.4 Operator Equations with Constraints ...................... 50
3.5 Mixed Formulations ..................................... 52
3.6 Coercive Operators ...................................... 57
4 Variational Formulations of Boundary Value Problems..... 59
4.1 Potential Equation ...................................... 59
4.1.1 Dirichlet Boundary Value Problem .................. 61
4.1.2 Lagrange Multiplier Methods ....................... 64
4.1.3 Neumann Boundary Value Problem.................. 67
4.1.4 Mixed Boundary Value Problem..................... 70
X Contents
4.1.5 Robin Boundary Value Problems .................... 71
4.2 Linear Elasticity ........................................ 72
4.2.1 Dirichlet Boundary Value Problem .................. 76
4.2.2 Neumann Boundary Value Problem.................. 77
4.2.3 Mixed Boundary Value Problems.................... 79
4.3 Stokes Problem ......................................... 79
4.4 Helmholtz Equation ..................................... 85
4.5 Exercises ............................................... 86
5 Fundamental Solutions .................................... 89
5.1 Laplace Operator........................................ 90
5.2 Linear Elasticity ........................................ 96
5.3 Stokes Problem .........................................101
5.4 Helmholtz Equation .....................................105
5.5 Exercises ...............................................109
6 Boundary Integral Operators ..............................111
6.1 Newton Potential........................................111
6.2 Single Layer Potential....................................118
6.3 Adjoint Double Layer Potential ...........................120
6.4 Double Layer Potential...................................124
6.5 Hypersingular Boundary Integral Operator .................128
6.6 Properties of Boundary Integral Operators..................136
6.6.1 Ellipticity of the Single Layer Potential ..............139
6.6.2 Ellipticity of the Hypersingular Boundary Integral
Operator .........................................144
6.6.3 Steklov–Poincar´e Operator .........................148
6.6.4 Contraction Estimates of the Double Layer Potential...149
6.6.5 Mapping Properties................................152
6.7 Linear Elasticity ........................................155
6.8 Stokes System ..........................................165
6.9 Helmholtz Equation .....................................167
6.10 Exercises ...............................................169
7 Boundary Integral Equations ..............................171
7.1 Dirichlet Boundary Value Problem.........................172
7.2 Neumann Boundary Value Problem........................175
7.3 Mixed Boundary Conditions ..............................179
7.4 Robin Boundary Conditions ..............................181
7.5 Exterior Boundary Value Problems ........................181
7.6 Helmholtz Equation .....................................183
7.7 Exercises ...............................................186
Contents XI
8 Approximation Methods...................................187
8.1 Galerkin–Bubnov Methods ...............................187
8.2 Approximation of the Linear Form.........................190
8.3 Approximation of the Operator ...........................191
8.4 Galerkin–Petrov Methods ................................193
8.5 Mixed Formulations .....................................195
8.6 Coercive Operators ......................................199
8.7 Exercises ...............................................202
9 Finite Elements............................................203
9.1 Reference Elements ......................................203
9.2 Form Functions .........................................211
9.3 Trial Spaces ............................................216
9.4 Quasi Interpolation Operators.............................225
9.5 Exercises ...............................................227
10 Boundary Elements........................................229
10.1 Reference Elements ......................................229
10.2 Trial Spaces ............................................233
11 Finite Element Methods ...................................243
11.1 Dirichlet Boundary Value Problem.........................243
11.2 Neumann Boundary Value Problem........................253
11.3 Finite Element Methods with Lagrange Multipliers ..........255
11.4 Exercises ...............................................261
12 Boundary Element Methods ...............................263
12.1 Dirichlet Boundary Value Problem.........................263
12.2 Neumann Boundary Value Problem........................274
12.3 Mixed Boundary Conditions ..............................281
12.4 Robin Boundary Conditions ..............................287
12.5 Exercises ...............................................289
13 Iterative Solution Methods ................................291
13.1 The Method of Conjugate Gradients .......................291
13.2 A General Preconditioning Strategy .......................299
13.2.1 An Application in Boundary Element Methods........302
13.2.2 A Multilevel Preconditioner
in Finite Element Methods .........................306
13.3 Solution Methods for Saddle Point Problems................319
14 Fast Boundary Element Methods ..........................327
14.1 Hierarchical Cluster Methods .............................328
14.2 Approximation of the Stiffness Matrix......................332
14.2.1 Taylor Series Representations .......................336
14.2.2 Series Representations of the Fundamental Solution....340
14.2.3 Adaptive Cross Approximation......................344
XII Contents
14.3 Wavelets ...............................................351
14.4 Exercises ...............................................366
15 Domain Decomposition Methods ..........................367
References.....................................................375
Index..........................................................383