Numerical Approximation of Partial Differential Equations

541 Pages·2016·4.98 MB·English

by  Sören Bartels

Texts in Applied Mathematics 64
Sören Bartels
Numerical
Approximation
of Partial
Diff erential
Equations
Texts in Applied Mathematics
Volume 64
Editors-in-chief:
StuartAntman,UniversityofMaryland,CollegePark,USA
LeslieGreengard,NewYorkUniversity,NewYorkCity,USA
PhilipHolmes,PrincetonUniversity,Princeton,USA
SeriesEditors:
JohnB.Bell,LawrenceBerkeleyNationalLab,Berkeley,USA
JosephB.Keller,StanfordUniversity,Stanford,USA
RobertKohn,NewYorkUniversity,NewYorkCity,USA
PaulNewton,UniversityofSouthernCalifornia,LosAngeles,USA
CharlesPeskin,NewYorkUniversity,NewYorkCity,USA
RobertPego,CarnegieMellonUniversity,Pittburgh,USA
LenyaRyzhik,StanfordUniversity,Stanford,USA
AmitSinger,PrincetonUniversity,Princeton,USA
AngelaStevens,UniversitätMünster,Münster,Germany
AndrewStuart,UniversityofWarwick,Coventry,UK
ThomasWitelski,DukeUniversity,Durham,USA
StephenWright,UniversityofWisconsin-Madison,Madison,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/1214
SoRren Bartels
Numerical Approximation
of Partial Differential
Equations
123
SoRrenBartels
AngewandteMathematik
Albert-Ludwigs-Universitaet
Freiburg,Germany
Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com.
ISSN0939-2475 ISSN2196-9949 (electronic)
TextsinAppliedMathematics
ISBN978-3-319-32353-4 ISBN978-3-319-32354-1 (eBook)
DOI10.1007/978-3-319-32354-1
LibraryofCongressControlNumber:2016940376
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Preface
Musicians are like mathematicians. Every part has to be right
forittowork,...
–JeffBeck,2015
This textbook is meant to serve as an introduction to the numerical analysis and
practical treatment of linear partial differential equations. It introduces the main
concepts for discretizing prototypical partial differential equations and discusses
theapplicationtomodelproblemsincontinuummechanicsandelectromagnetism.
ShortMATLABimplementationsillustratethepracticalityofthenumericalmethods.
Thefirstpartofthetextbookcoversthedevelopmentandanalysisofelementary
finite difference methods, the mathematical theory for elliptic partial differential
equations, and the constructionand numericalanalysis of finite elementmethods.
The second part is devoted to iterative solution methods either by refining a
triangulation locally to efficiently resolve corner singularities, or by solving the
resulting linear system of equations via multilevel or domain decomposition
techniques.Inthethirdpartofthetext,linearlyconstrainedandsingularlyperturbed
boundaryvalueproblemsareinvestigated.Theirnumericaltreatmentisbasedona
mathematicalframeworkforsaddle-pointproblemsandrequiresusingnonstandard
finite element spaces. This leads to accurate numerical schemes for simulating
nearly incompressible materials, thin elastic objects, electromagnetic fields, and
turbulentflows.
The reader is assumed to be familiar with basic numerical techniques such
as interpolation, quadrature, sparse matrices, and iterative solution techniques for
linear systems of equations, although this is not mandatory for understandingthe
mainconcepts.Itisalsohelpfultohaveexperiencewithlinearfunctionalanalysis,
propertiesofLebesgueandSobolevspaces,andtheoriesofexistenceandregularity
forlinearpartialdifferentialequations.Thecorrespondingresultsneededinthistext
areexplainedbutnotprovedindetail.
Each of the three parts of the book can be used for a one-term lecture accom-
paniedbytheoreticalandpracticaltutorials.Theproblemsandprojectsincludedin
thetextmayrequireadditionalcommentsandsomespecificsupervisiondepending
v
vi Preface
on the prerequisites of the students. Their main purpose is to provide ideas for
individualworkandexperimentswiththeincludedMATLABcodesthatareavailable
at
http://extras.springer.com/2016/978-3-319-32353-4
In Chap.1 we discussthe mathematicaldescriptionoftransport,diffusion,and
wave phenomena and their numerical simulation with finite difference methods.
Weinvestigatetheaccuracyofthemethodsviastabilityandconsistencyproperties
assuming the existence of regular solutions. Optimal order convergence rates for
generalboundaryconditionswillbe addressedand thepracticalityofthe methods
illustratedwithshortimplementations.
Chapter 2 is concerned with general existence theories for solutions of partial
differential equations using concepts from functional analysis and considering
generalizations of classical derivatives based on a multidimensional integration-
by-parts formula. The chapter introduces Sobolev spaces, discusses their main
properties, states existence theories for elliptic second order linear partial differ-
entialequations,andsketchesregularityresultsforsolutions.
InChap.3weshowhowfiniteelementmethodsprovideanabstractframework
forinterpolatingfunctionsorvectorfieldsinmultidimensionaldomains.Theyallow
forspecifyingGalerkinmethodsforapproximatingpartialdifferentialequations.In
combinationwithregularityresults,errorestimatesinvariousnormscanbeproved.
Wediscusstheefficientimplementationofloworderandisoparametricmethodsin
thecaseofstationaryandevolutionarymodelproblems.
The starting point for Chap. 4 is the observation that convergence rates of
standard numerical methods are suboptimal when the solution has corner singu-
larities as in the case of elliptic equations on nonconvex domains. Optimal rates
canbeobtainedbyusinglocallyrefinedtriangulationswhichareeitherspecifically
constructed for particular domains or generated automatically via adaptive mesh-
refinement algorithms. We introduce both approaches, analyze their convergence,
andillustratetheirimplementation.
InChap.5wemakeuseofthefactthatlinearsystemsofequationsresultingfrom
finite element discretizations of partial differential equations are typically large,
sparse, and ill-conditioned. Their efficient numerical solution exploits properties
oftheunderlyingcontinuousproblemorasequenceofdiscretizations.Thechapter
discussesmultigrid,domaindecomposition,andpreconditioningmethods.
ThegoalofChap.6istoprovidealternativestostandardnumericalmethodsthat
failtoprovideaccurateapproximationswhen partialdifferentialequationsinvolve
constraintsdefinedbya differentialoperatororwhentheycontaintermsweighted
byalargerparameter.GeneralizationsoftheLax–MilgramandCéalemmasprovide
a concise framework for the development and analysis of appropriate numerical
methods. Central to the construction is the validity of an inf-sup condition that
definesacompatibilityrequirementoninvolvedfiniteelementspaces.
In Chap. 7 we introduce and analyze stable finite element methods for dis-
cretizingasaddle-pointformulationofthePoissonproblemandtheStokessystem.
Moreover, we investigate characteristic properties of convection-dominatedequa-
Preface vii
tionsandtheirnumericalapproximationviaintroducingstabilizingterms.Flexible
discontinuous Galerkin methods are derived and analyzed for a model Poisson
equation. Simple implementations illustrate the performance of the numerical
methods.
The final Chap. 8 discusses the development, analysis, and implementation of
numerical methods for boundary value problems in elasticity, electromagnetism,
andfluidmechanics.Eachoftheconsideredproblemsrequiresasuitablenumerical
treatmenttocapturerelevanteffectswithalownumberofdegreesoffreedom.
ThistextbookresultsfromseveralcourseswhichIhavetaughtattheUniversity
of Maryland at College Park, the Humboldt University of Berlin, the University
of Bonn, and the Albert Ludwig University Freiburg. Besides many colleagues,
assistants,andstudentsthathavecontributedtothedevelopmentofthistext,Iwould
particularlyliketothankMarijoMilicevic,AlexisPapathanassopoulos,DirkPauly,
andPatrickSchönfortheirhelpandsupport.
Freiburg,Germany SörenBartels
March2015

Contents
PartI FiniteDifferencesandFiniteElements
1 FiniteDifferenceMethod.................................................... 3
1.1 TransportEquation..................................................... 3
1.1.1 MathematicalModel .......................................... 3
1.1.2 ExplicitSolution............................................... 4
1.1.3 DifferenceQuotients .......................................... 5
1.1.4 ApproximationScheme ....................................... 7
1.1.5 Stability ........................................................ 9
1.1.6 Convergence................................................... 11
1.1.7 CFLCondition................................................. 12
1.1.8 FourierStabilityAnalysis..................................... 13
1.2 HeatEquation........................................................... 15
1.2.1 MathematicalModel .......................................... 15
1.2.2 ExplicitSolution............................................... 17
1.2.3 PropertiesofSolutions........................................ 18
1.2.4 ExplicitScheme ............................................... 20
1.2.5 ImplicitScheme ............................................... 22
1.2.6 MidpointScheme.............................................. 26
1.2.7 Crank–NicolsonScheme...................................... 29
1.2.8 SourceTermsandBoundaryConditions..................... 33
1.3 WaveEquation.......................................................... 34
1.3.1 MathematicalModel .......................................... 34
1.3.2 ExplicitSolution............................................... 36
1.3.3 Well-Posedness................................................ 38
1.3.4 ExplicitScheme ............................................... 39
1.3.5 ImplicitScheme ............................................... 42
1.4 PoissonEquation....................................................... 44
1.4.1 MathematicalModel .......................................... 44
1.4.2 PoissonProblem............................................... 46
1.4.3 FiniteDifferenceScheme ..................................... 49
ix