Numerical approximation of partial differential equations

544 Pages·1994·0.4 MB·English

by  Alfio Quarteroni

S 23
pringer Series in
C
omputational
M
athematics
EditorialBoard
R.L.Graham
J.Stoer
R.Varga
Alfio Quarteroni · Alberto Valli
Numerical Approximation
of Partial Differential
Equations
With59Figuresand17Tables
123
AlfioQuarteroni AlbertoValli
ÉcolePolytechniqueFédérale UniversitàdiTrento
deLausanne DipartimentodiMatematica
ChairedeModelisationet ViaSommarive,14
CalculScientifique(CMCS) 38050PovoTN,Italy
Station8 [email protected]
1015Lausanne,Switzerland
alfio.quarteroni@epfl.ch
and
MOX,PolitecnicodiMilano
20133Milan,Italy
Firstsoftcoverprinting2008
ISBN 978-3-540-85267-4 e-ISBN 978-3-540-85268-1
DOI 10.1007/978-3-540-85268-1
SpringerSeriesinComputationalMathematicsISSN0179-3632
LibraryofCongressControlNumber:97160884
MathematicsSubjectClassification(1991):65Mxx,65Nxx,65Dxx,65Fxx,35Jxx,35Kxx,
35Lxx,35Q30,76Mxx
©2008,1994 Springer-VerlagBerlinHeidelberg
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthe
materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,
recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindata
banks.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisions
oftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermission
forusemustalwaysbeobtainedfromSpringer.Violationsareliabletoprosecutionunderthe
GermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
Coverdesign:WMXDesignGmbH,Heidelberg
Typesetting:bytheauthorsusingaSpringerTEXmacropackage
Printedonacid-freepaper
98765 4321
springer.com
A Fulvia e Tiziana
Preface
Everything is more simple than one thinks
but at the same time more complex than one can understand
Johann Wolfgang von Goethe
To reach the point that is unknown to you,
you must take the road that is unknown to you
St. John of the Cross
Thisisa bookon thenumericalapproximationofpartialdifferentialequations
(PDEs). Its scope is to provide a thorough illustration ofnumerical methods
(especially those stemming from the variationalformulation ofPDEs), carry
out their stability and convergenceanalysis, derive error bounds, and discuss
the algorithmic aspects relative to their implementation.
A sound balancing of theoretical analysis, description of algorithms and
discussion ofapplications is our primary concern.
Many kinds of problems are addressed: linear and nonlinear, steady
and time-dependent, having either smooth or non-smooth solutions. Besides
model equations, weconsider a number of(initial-) boundaryvalue problems
of interest in several fields of applications.
Part I is devoted to the description and analysis of general numerical
methods for the discretization of partial differential equations.
A comprehensive theory of Galerkin methods and its variants (Petrov
GalerkinandgeneralizedGalerkin),aswellasofcollocationmethods,isdevel
opedfor thespatialdiscretization. This theoryisthenspecified to twonumer
ical subspace realizations of remarkable interest: the finite element method
(conforming, non-conforming, mixed, hybrid) and the spectral method (Leg
endre and Chebyshev expansion).
For unsteady problems we will illustrate finite difference and fractional
step schemes for marching in time. Finite differences will also be extensively
considered in Parts II and IIIin the framework ofconvection-diffusion prob
lems and hyperbolicequations. For the latterwewillalso address, briefly,the
schemes based on finite volumes.
For the solution of algebraic systems, which are typically very large and
sparse, we revise classical and modern techniques, either direct and iterative
with preconditioning, for both symmetric and non-symmetric matrices. A
VIII Preface
short account will be given also to multi-grid and domain decomposition
methods.
Parts II and III are respectively devoted to steady and unsteady prob
lems. For each (initial-) boundary value problem we consider, we illustrate
the main theoretical results about well-posedness, i.e., concerning existence,
uniqueness and a-prioriestimates. Afterwards, wereconsiderand analyzethe
previously mentioned numerical methods for the problem at hand, wederive
the corresponding algebraic formulation, and we comment on the solution
algorithms.
Tobegin with, weconsiderallclassical equationsofmathematicalphysics:
elliptic equations for potential problems, parabolic equations for heat diffu
sion, hyperbolicequationsforwavepropagationphenomena.Furthermore,we
discuss extensively advection-diffusion equations for passive scalars and the
Navier-Stokes equations (together with their linearized version, the Stokes
problem) for viscous incompressible flows. We also derive the equations of
fluid dynamics in their general form.
Unfortunately, the limitation of space and our own experience have re
sulted in the omission ofmany important topics that wewould have liked to
include (for example, the Saint-Venant model for shallow water equations,
the system of linear elasticity and the biharmonic equation for membrane
displacement and thin plate bending, the drift-diffusion and hydrodynamic
models for semiconductor devices, the Navier-Stokesand Eulerequationsfor
compressible flows).
This book is addressed to graduatestudents as wellas to researchersand
specialistsin thefieldofnumericalsimulationofpartialdifferentialequations.
As a graduate text for Ph.D. courses it may be used in its entirety. Part
I may be regarded as a one quarter introductory course on variational nu
merical methods for PDEs. Part II and III deal with its application to the
numerical approximationoftime-independent and time-dependent problems,
respectively, and could be taught through the two remaining quarters. How
ever, other solutions may work well.For instance, supplementingPart I with
Chapters 6, 11and most partof14may besuitableforaone semestercourse.
The rest of the book could be covered in the second semester. Following a
different key, Part I plus Chapters 8, 9, 10, 12, 13 and 14 can be regarded
as an introduction to numerical fluid dynamics. Other combinations are also
envisageable.
The authors are grateful to Drs. C. Byrne and J. Heinze of Springer
Verlag for their encouragement throughout this project. The assistence of
the technical staffof Springer-Verlag has contributed to the final shaping of
.the manuscript.
This book benefitsfrom our experiencein teachingthesesubjectsoverthe
past years in different academical institutions (the University of Minnesota
at Minneapolis, the Catholic University of Brescia and the Polythecnic of
Milan for the first author, the University of Trento for the second author),
Preface IX
and from students' reactions. Help was given to us by several friends and
collaborators who read parts ofthe manuscript or provided figures or tables.
In this connection weare happyto thankV.1.Agoshkov,Yu.A.Kuznetsov,D.
Ambrosi,L. Bergamaschi,S.Delladio, M.Manzini,M.Paolini,F. Pasquarelli,
L. Stolcis, E. Zampieri, A. Zaretti and in particular C. Bernini, P. Gervasio
and F. Saleri.
We would also wish to thank Ms. R. Holliday for having edited the lan
guage ofthe entire manuscript. Finally, the expert and incredibly adept typ
ing of the lEX-files by Ms. C. Foglia has been invaluable.
Milan and Trento AlfioQuarteroni
May, 1994 Alberto Valli
In the second printing of this book we have corrected several misprints,
and introduced some modifications to the original text.
More precisely, we have sligthly changed Sections 2.3.4, 3.4.1, 8.4 and
12.3, and we have added some further comments to Remark 8.2.1.
We have also completed the references of those papers appeared after
1994.
Milan and Trento AlfioQuarteroni
December, 1996 AlbertoValli
Table of Contents
Part I. Basic Concepts and Methods for PDEs' Approximation
1. Introduction 1
1.1 The Conceptual Path Behind the Approximation 2
1.2 Preliminary Notation and Function Spaces 4
1.3 Some Results About Sobolev Spaces 10
1.4 Comparison Results 13
2. Numerical Solution ofLinear Systems 17
2.1 Direct Methods 17
2.1.1 Banded Systems 22
2.1.2 Error Analysis 23
2.2 Generalities on Iterative Methods 26
2.3 Classical Iterative Methods 29
2.3.1 Jacobi Method 29
2.3.2 Gauss-Seidel Method 31
2.3.3 Relaxation Methods (S.O.R. and S.S.O.R.) . 32
2.3.4 Chebyshev Acceleration Method 34
2.3.5 The Alternating Direction Iterative Method 37
2.4 Modern Iterative Methods 39
2.4.1 Preconditioned Richardson Method 39
2.4.2 Conjugate Gradient Method 46
2.5 Preconditioning 51
2.6 Conjugate Gradient and Lanczos like Methods for
Non-Symmetric Problems 57
2.6.1 GCR, Orthomin and Orthodir Iterations .......... 57
2.6.2 Arnoldi and GMRES Iterations 59
2.6.3 Bi-CG, CGS and Bi-CGSTAB Iterations 62
2.7 The Multi-Grid Method 65
2.7.1 The Multi-Grid Cycles 65
2.7.2 A Simple Example 67
2.7.3 Convergence 70
2.8 Complements 71
XII Table ofContents
3. Finite Element Approximation 73
3.1 Triangulation 73
3.2 Piecewise-Polynomial Subspaces 74
3.2.1 The Scalar Case 75
3.2.2 The Vector Case 76
3.3 Degrees of Freedom and Shape Functions 77
3.3.1 The Scalar Case: Triangular Finite Elements 77
3.3.2 The Scalar Case: Parallelepipedal Finite Elements .. 80
3.3.3 The Vector Case 82
3.4 The Interpolation Operator 85
3.4.1 Interpolation Error: the Scalar Case 85
3.4.2 Interpolation Error: the Vector Case 91
3.5 Projection Operators 96
3.6 Complements 99
4. Polynomial Approximation 101
4.1 Orthogonal Polynomials 101
4.2 Gaussian Quadrature and Interpolation 103
4.3 Chebyshev Expansion 105
4.3.1 Chebyshev Polynomials 105
4.3.2 Chebyshev Interpolation 107
4.3.3 Chebyshev Projections 113
4.4 Legendre Expansion 115
4.4.1 Legendre Polynomials 115
4.4.2 Legendre Interpolation 117
4.4.3 Legendre Projections 120
4.5 Two-Dimensional Extensions 121
4.5.1 The Chebyshev Case 121
4.5.2 The Legendre Case 124
4.6 Complements 127
5. Galerkin, Collocation and Other Methods 129
5.1 An Abstract Reference Boundary Value Problem 129
5.1.1 Some Results of Functional Analysis 133
5.2 Galerkin Method 136
5.3 Petrov-Galerkin Method 138
5.4 Collocation Method 140
5.5 Generalized Galerkin Method 141
5.6 Time-Advancing Methods for Time-Dependent Problems 144
5.6.1 Semi-Discrete Approximation 148
5.6.2 Fully-Discrete Approximation 148
5.7 Fractional-Step and Operator-Splitting Methods 151
5.8 Complements 156
TableofContents XIII
Part II. Approximation of Boundary Value Problems
6. Elliptic Problems: Approximation by Galerkin and
Collocation Methods 159
6.1 Problem Formulation and Mathematical Properties 159
6.1.1 Variational Form ofBoundary Value Problems 161
6.1.2 Existence, Uniqueness and A-Priori Estimates 164
6.1.3 Regularity of Solutions 167
6.1.4 On the Degeneracy of the Constants in Stability
and Error Estimates 168
6.2 Numerical Methods: Construction and Analysis 169
6.2.1 Galerkin Method: Finite Element and Spectral
Approximations 170
6.2.2 Spectral Collocation Method 179
6.2.3 Generalized Galerkin Method 187
6.3 Algorithmic Aspects 189
6.3.1 Algebraic Formulation 190
6.3.2 The Finite Element Case 192
6.3.3 The Spectral Collocation Case 198
6.4 Domain Decomposition Methods 204
6.4.1 The Schwarz Method 206
6.4.2 Iteration-by-Subdomain Methods Based on
Transmission Conditions at the Interface 209
6.4.3 The Steklov-Poincare Operator 212
6.4.4 The Connection Between Iterations-by-Subdomain
Methods and the Schur Complement System 215
7. Elliptic Problems: Approximation by Mixed and
Hybrid Methods 217
7.1 Alternative Mathematical Formulations 217
7.1.1 The Minimum Complementary Energy Principle 218
7.1.2 Saddle-Point Formulations: Mixed and Hybrid
Methods 222
7.2 Approximation by Mixed Methods 230
7.2.1 Setting up and Analysis 230
7.2.2 An Example: the Raviart-Thomas Finite Elements .. 235
7.3 Some Remarks on the Algorithmic Aspects 241
7.4 The Approximation of More General Constrained
Problems 246
7.4.1 Abstract Formulation 246
7.4.2 Analysis of Stability and Convergence 250
7.4.3 Howto Verifythe Uniform Compatibility Condition 253
7.5 Complements 255