Partial Differential Equations: Modeling, Analysis and Numerical Approximation

403 Pages·2016·4.07 MB·English

by  Hervé Le Dret

International Series of Numerical Mathematics
168
Hervé Le Dret
Brigitte Lucquin
Partial Differential
Equations:
Modeling, Analysis
and Numerical
Approximation

ISNM
International Series of Numerical Mathematics
Volume 168
Managing Editor
G. Leugering, Erlangen-Nürnberg, Germany
Associate Editors
Z. Chen, Beijing, China
R.H.W. Hoppe, Augsburg, Germany; Houston, USA
N. Kenmochi, Chiba, Japan
V. Starovoitov, Novosibirsk, Russia
Honorary Editor
K.-H. Hoffmann, München, Germany
More information about this series at www.birkhauser-science.com/series/4819
é
Herv Le Dret Brigitte Lucquin
(cid:129)
Partial Differential Equations:
Modeling, Analysis and
Numerical Approximation
HervéLeDret Brigitte Lucquin
Laboratoire Jacques-Louis Lions Laboratoire Jacques-Louis Lions
UniversitéPierre etMarie Curie—Paris VI UniversitéPierre etMarie Curie—Paris VI
Paris Paris
France France
ISSN 0373-3149 ISSN 2296-6072 (electronic)
International Series of Numerical Mathematics
ISBN978-3-319-27065-4 ISBN978-3-319-27067-8 (eBook)
DOI 10.1007/978-3-319-27067-8
LibraryofCongressControlNumber:2015955864
Mathematics Subject Classification (2010): 35J20, 35J25, 35K05, 35K20, 35L03, 35L05, 65M06,
65M08,65M12,65N30
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Thisbookisdevotedtothestudyofpartialdifferentialequationproblemsbothfrom
thetheoreticalandnumericalpointsofview.Apartialdifferentialequation(PDE)is
a relation between the partial derivatives of an unknown function u in several
variables to be satisfied by this function, for example ou(cid:1)ou ¼ 0, where u is a
ot ox
functioninthetwovariablestandx.Partialdifferentialequationsconstituteamajor
field of study in contemporary mathematics. They also arise in other fields of
mathematics, such as differential geometry or probability theory for example. In
addition, partialdifferentialequationsappearinawidevarietyofcontextsinmany
applied fields, not only in the traditional fields of physics, mechanics and engi-
neering, but also more recently in chemistry, bioscience, medicine, meteorology,
climatology, finance and so on.
In all of these applied fields, numerical simulation is playing an increasingly
prominentrole,becauseeventhoughsolutionstosuchPDEproblemscanbeshown
to exist, there is in general no closed form solution. Therefore, quantitative infor-
mation about the solutions can only be obtained by means of numerical approxi-
mationmethods.Itisveryimportantnottotakenumericalsimulationresultsatface
value since they inherently present errors. In order to be able to do this, a com-
prehensiveknowledgeofallthestepsinvolvedisnecessary,startingfrommodeling
to mathematical existence theorems, to numerical approximation methods. This is
one of the main purposes of this book.
A numerical approximation method is a procedure in which the original con-
tinuous unknowns are replaced by a finite number of discrete, computable
approximateunknowns.Itisthusimportanttobeabletoproperlyunderstandonthe
one hand the properties of partial differential equations and, on the other hand, the
propertiesofthenumericalmethodsthatareusedtodefineapproximationsoftheir
solutionsandeffectively compute such approximations. Inparticular,itisessential
to quantify the approximation by appropriately defining the error between
approximate and continuous unknowns, and to show that this error goes to zero,
preferably at a known rate, in the continuous limit when the number of approxi-
mated unknowns goes to infinity.
v
vi Preface
The goal of this book is to try and illustrate this program through a rather wide
spectrumofclassicalornotsoclassicalexamples.Wethuspresentsomemodeling
aspects,developthetheoreticalanalysisofthepartialdifferentialequationproblems
thusobtainedforthethreemainclassesofpartialdifferentialequations,andanalyze
several numerical approximation methods adapted to each of these examples.
We have selected three broad families of numerical methods, finite difference,
finite element and finite volumes methods. This is far from being exhaustive, but
these three families of methods are the most widely used and constitute the core
skillsforanyoneintendingtoworkwithnumericalsimulationofpartialdifferential
equations. There are many other numerical methods that we have chosen not to
develop, in order to keep the size of the book within reasonable bounds.
Parts of the book are accessible to Bachelor students in mathematics or engi-
neering. However, most of the book is better suited to Master students in applied
mathematics or computational engineering. We put emphasis on mathematical
detail and rigor for the analysis of both continuous and discrete problems.
The book is structured globally according to the three major types of partial
differential equations: elliptic equations, parabolic equations and hyperbolic equa-
tions. We mainly consider linear equations, except for a few nonlinear examples.
Each one of the above three types of equations requires a specific set of mathe-
maticaltechniquesfortheoreticalanalysis,andaspecificsetofnumericalmethods,
i.e.,specificdiscretizationproceduresandconvergenceanalysesforapproximation.
We follow a path of progressive difficulty in each case inasmuch as possible. We
begin with the most elementary approaches either theoretical or numerical, which
alsohappentobetheearliestonesfromthehistoricalviewpoint.Wethencontinue
to more advanced topics that require a more sophisticated mathematical back-
ground, both from the theoretical and numerical points of view, and that are also
more recent than the previous ones. We also give along theway severalnumerical
illustrations of successes as well as failures of numerical methods, using free
software such as Scilab and FreeFem++.
The book is divided into ten chapters. Chapter 1 is devoted to mathematical
modeling. We give examples ranging from mechanics and physics to finance.
Starting from concrete situations, we try to present the various steps leading to a
mathematical model involving partial differential equations to the extent possible.
In some cases, it is possible to start from first principles and entirely derive the
equationsandboundaryconditions.Inother,morecomplicatedcases,wejustgivea
few indications concerning the modeling approach and sometimes a historical
account. For some of the examples considered, we also give a short elementary
mathematical study: existence, uniqueness, maximum principle.
In Chap. 2, we present the simplest possible and earliest numerical method, the
so-called finite difference method. We first use the example of a one-dimensional
elliptic problem,alreadyintroducedinChap.1,whichiselementaryenoughnotto
require sophisticated mathematical machinery. We then consider a few general-
izations: one-dimensional problems with other boundary conditions, two-
dimensional problems, still on a rather elementary mathematical level.
Preface vii
ThefirsttwochaptersareaccessibleattheBachelorlevel.Theremainder ofthe
bookcallsforahigherlevelofmathematics,withthepossibleexceptionofpartsof
Chaps. 8 and 10. Chapter 3 is thus devised as a toolbox of more advanced math-
ematical analysis techniques that are required to study more general partial differ-
ential equations, especially in more than one dimension: a review of Hilbert space
theory, usual function spaces, properties of open sets in Rd, multidimensional
integrationbyparts,distributions,andSobolevspaces.Theresultsarestandardand
we sometimes refer to classical references. We have however detailed some of the
proofs for those results that are not always easy to find in the literature. Readers
who are more interested in the numerical aspects can leaf through this chapter and
keepit for future reference. Forthereader’sconvenience,thechapterisconcluded
by a summary of the most important results that are used in the sequel.
Chapter 4 is concerned with the variational formulation of multidimensional
ellipticboundaryvalueproblems,whichisaverypowerfulwayofrewritingcertain
partial differential equation problems in an abstract Hilbertian setting, in order to
prove the existence and uniqueness of solutions. We provide many examples of
such problems. Most of these examples are problems of second order, i.e., the
maximum order of derivatives appearing in the partial differential equation is two,
withonefourthorderexample.Wealsoconsideravarietyofboundaryconditions.
The variational formulation also makes it possible to devise numerical approx-
imationmethodsinaverynaturalandunifiedfashion.Weintroducesuchmethods
in Chap. 5. The approximate problems are set in the same framework as the con-
tinuous problem. This framework also provides a fundamental error estimate.
Of particular interest to usis thefinite element method introduced and analyzedin
detail on simple one-dimensional examples.
In Chap. 6, we study the generalization of the finite element method to
two-dimensional elliptic problems. We start by a detailed presentation of approx-
imationusingrectangularfiniteelements.Weprovideaconvergenceestimateinthe
case of the Lagrange rectangular element of lowest possible degree and give
indicationsaboutconvergenceforhigherdegrees.Wethenintroducetheconceptof
barycentric coordinates and use them to describe Lagrange triangular finite ele-
ments of degree up to three.
The elliptic problems considered so far correspond to the modeling of static or
equilibriumsituations,withnotimeevolution.Wethenturntoevolutionproblems
inChap.7withthetheoreticalstudyoftheheatequation,whichistheprototypical
parabolic equation: maximum principle, existence and uniqueness of regular
solutions, energy estimates, variational formulation and weak solutions. New
mathematicaltoolsareneeded,mainlyHilbertspace-valuedfunctionspaces,which
weintroduceastheneedarises.Wealsomentionanddiscusstheheatkernelinthe
case of the heat equation in the whole space.
We next consider the numerical approximation of the heat equation in Chap. 8.
We focus on the finite difference method, already seen for static problems in
Chap. 2. We consider several finite difference schemes of various precision. The
convergence of such schemes rest on their consistency and stability. The analysis
viii Preface
oftheseschemesissignificantlymorecomplicatedintheevolutioncasethaninthe
static case, due in particular to rather subtle stability issues, which we analyze in
detail and from several different viewpoints. We also mention other methods such
as the finite difference-finite element method, in which time is discretized using
finite differences and space is discretized using finite elements.
Chapter 9 is devoted to both theoretical and numerical analyses of another
classical evolution problem, the wave equation. This equation is the prototypical
hyperbolic equation of second order. We first study the continuous problem and
provetheexistenceanduniquenessofregularsolutionsandthenofweaksolutions.
We next consider finite difference schemes for the wave equation. The stability
issues areagain significantlysubtlerthan inthecase oftheheat equation, andtake
up most of the exposition concerning numerical methods.
Finally, we present the finite volume method in Chap. 10, again on examples.
Finitevolumemethodsarethemostrecentofthenumericalmethodscoveredinthe
book. They are currently widely used in certain areas of applications such as
computational fluid dynamics. We start with the one-dimensional elliptic problem
of Chap. 2 with a description of the finite volume discretization and a complete
convergence analysis. We then consider the one-dimensional transport equation,
whichistheprototypicalhyperbolicequationoffirstorder.Thisequationissolved
via the method of characteristics. We then introduce several linear finite volume
schemes and study their properties of consistency, stability and convergence. We
also present the method on a few examples in the nonlinear case and for the
two-dimensional transport equation.
The contents of this book are significantly expanded from a series offirst year
Master degree classes, taught by the authors at UPMC (Université Pierre et Marie
Curie) in Paris, France, over several years. It is intended to be as self-contained as
possible. It consequently provides more than enough material for a one semester
Masterclassonthesubject.Thebookcanalsoserveasawidespectrummonograph
for more advanced readers.
We areindebted tothe students who have attended our classes andaskedmany
questionsthatcontributedtomakingthetextmorereadable.Wewouldliketothank
our colleagues Muriel Boulakia, Edwige Godlewski, Sidi Mahmoud Kaber and
Gérard Tronel, for carefullyreading parts of the manuscript and making numerous
suggestions that improved it significantly. We also thank the anonymous referees
for making several constructive comments.
Paris, France Hervé Le Dret
Brigitte Lucquin
Contents
1 Mathematical Modeling and PDEs . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Elastic String. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Elastic Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 The Elastic Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 The Transport Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 The Vibrating String Equation. . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7 The Heat Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 The Schrödinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.9 The Black and Scholes Equation . . . . . . . . . . . . . . . . . . . . . . 31
1.10 A Rough Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . 33
1.11 What Comes Next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 The Finite Difference Method for Elliptic Problems. . . . . . . . . . . . 35
2.1 Approximating Derivatives by Differential Quotients . . . . . . . . 35
2.2 Application to a One-Dimensional Model Problem. . . . . . . . . . 38
2.3 Convergence of the Finite Difference Method . . . . . . . . . . . . . 41
2.4 Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 The Two-Dimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 A Review of Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1 Basic Hilbert Space Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 A Few Basic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3 Regularity of Open Subsets of Rd . . . . . . . . . . . . . . . . . . . . . 76
3.4 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Integration by Parts in Dimension d and Applications. . . . . . . . 87
3.6 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.7 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
ix