The Mathematical Theory of Finite Element Methods

404 Pages·1994·2.145 MB·English

by  Susanne C. Brenner, L. Ridgway Scott (auth.)

15
Texts in Applied Mathematics
Editors
J.E.Marsden
L.Sirovich
S.S.Antman
Advisors
G.Iooss
P.Holmes
D.Barkley
M.Dellnitz
P.Newton
Texts in Applied Mathematics
1. Sirovich:IntroductiontoAppliedMathematics.
2. Wiggins:IntroductiontoAppliedNonlinearDynamicalSystemsandChaos.
3. Hale/Koc¸ak:DynamicsandBifurcations.
4. Chorin/Marsden:AMathematicalIntroductiontoFluidMechanics,3rded.
5. Hubbard/West:DifferentialEquations:ADynamicalSystemsApproach:
OrdinaryDifferentialEquations.
6. Sontag:MathematicalControlTheory:DeterministicFiniteDimensional
Systems,2nded.
7. Perko:DifferentialEquationsandDynamicalSystems,3rded.
8. Seaborn:HypergeometricFunctionsandTheirApplications.
9. Pipkin:ACourseonIntegralEquations.
10. Hoppensteadt/Peskin:ModelingandSimulationinMedicineandtheLife
Sciences,2nded.
11. Braun:DifferentialEquationsandTheirApplications,4thed.
12. Stoer/Bulirsch:IntroductiontoNumericalAnalysis,3rded.
13. Renardy/Rogers:AnIntroductiontoPartialDifferentialEquations.
14. Banks:GrowthandDiffusionPhenomena:MathematicalFrameworksand
Applications.
15. Brenner/Scott:TheMathematicalTheoryofFiniteElementMethods,3rded.
16. VandeVelde:ConcurrentScientificComputing.
17. Marsden/Ratiu:IntroductiontoMechanicsandSymmetry,2nded.
18. Hubbard/West:DifferentialEquations:ADynamicalSystemsApproach:
Higher-DimensionalSystems.
19. Kaplan/Glass:UnderstandingNonlinearDynamics.
20. Holmes:IntroductiontoPerturbationMethods.
21. Curtain/Zwart:AnIntroductiontoInfinite-DimensionalLinearSystems
Theory.
22. Thomas:NumericalPartialDifferentialEquations:FiniteDifference
Methods.
23. Taylor:PartialDifferentialEquations:BasicTheory.
24. Merkin:IntroductiontotheTheoryofStabilityofMotion.
25. Naber:Topology,Geometry,andGaugeFields:Foundations.
26. Polderman/Willems:IntroductiontoMathematicalSystemsTheory:
ABehavioralApproach.
27. Reddy:IntroductoryFunctionalAnalysiswithApplicationstoBoundary
ValueProblemsandFiniteElements.
28. Gustafson/Wilcox:AnalyticalandComputationalMethodsofAdvanced
EngineeringMathematics.
29. Tveito/Winther:IntroductiontoPartialDifferentialEquations:
AComputationalApproach
(continuedafterindex)
Susanne C. Brenner L. Ridgway Scott
The Mathematical Theory
of Finite Element Methods
Third Edition
ABC
SusanneC.Brenner L.RidgwayScott
DepartmentofMathematicsandCenter UniversityofChicago
forComputationandTechnology Chicago,IL60637
LouisianaStateUniversity USA
BatonRouge,LA70803 [email protected]
USA
[email protected]
SeriesEditors
L.Sirovich
J.E.Marsden DivisionofAppliedMathematics
ControlandDynamicSystems,107-81 BrownUniversity
CaliforniaInstituteofTechnology Providence,RI02912
Pasadena,CA91125 USA
USA
S.S.Antman
DepartmentofMathematics
and
InstituteforPhysicalScience
andTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
ISBN978-0-387-75933-3 e-ISBN978-0-387-75934-0
DOI:10.1007/978-0-387-75934-0
LibraryofCongressControlNumber:2007939977
MathematicsSubjectClassification(2000):65N30,65–01,46N40,65M60,74S05
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
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Series Preface
Mathematics is playing an ever more important role in the physical and
biological sciences, provoking a blurring of boundaries between scientific
disciplines and a resurgence of interest in the modern as well as the clas-
sical techniques of applied mathematics. This renewal of interest, both in
research and teaching, has led to the establishment of the series Texts in
Applied Mathematics (TAM).
The development of new courses is a natural consequence of a high
level of excitement on the research frontier as newer techniques, such as
numerical and symbolic computer systems, dynamical systems, and chaos,
mix with and reinforce the traditional methods of applied mathematics.
Thus, the purpose of this textbook series is to meet the current and future
needs of these advances and to encourage the teaching of new courses.
TAMwillpublishtextbookssuitableforuseinadvancedundergraduate
and beginning graduate courses, and will complement the Applied Mathe-
maticalSciences(AMS)series,whichwillfocusonadvancedtextbooksand
research-level monographs.
Pasadena, California J.E. Marsden
Providence, Rhode Island L. Sirovich
College Park, Maryland S.S. Antman
Preface to the Third Edition
This edition contains four new sections on the following topics: the BDDC
domain decomposition preconditioner (Section 7.8), a convergent adap-
tive algorithm (Section 9.5), interior penalty methods (Section 10.5) and
Poincar´e-Friedrichs inequalities for piecewise W1 functions (Section 10.6).
p
We have made improvements throughout the text, many of which were
suggested by colleagues, to whom we are grateful. New exercises have been
added and the list of references has also been expanded and updated.
Some of the new material originated from our research and we would
like to thank the National Science Foundation for support. The first au-
thor would also like to thank the Alexander von Humboldt Foundation for
supporting her visit to Germany in the Summer of 2007, during which the
work on this edition was completed. The second author would also like to
thanktheUniversit´ePierreetMarieCurieforsupportinghisvisitstoParis
duringthepastseveralyears,duringwhichworkrelatedtothiseditionwas
carried out.
In the preface to the first edition, we outlined different ways the book
couldbeusedincourses,butsincesomechapternumbershavechanged,we
rephrase these suggestions here. Chapters 0 through 5 form the essential
material for a course (these chapter numbers have not changed). Chapters
6 and 7 provide an introduction to efficient iterative solvers for the linear
systems of finite element equations, but they do not contain material re-
quired by later chapters. A course emphasizing algorithmic aspects would
includethem.Similarly,Chapters8and9arenotrequiredinlaterchapters.
Acoursecoveringchallenginganalysisquestionswouldcoverthese.Thefor-
mer develops and applies max-norm error estimates to nonlinear problems,
andthelatterintroducestheconceptofmeshadaptivity.Chapter10,how-
ever, has an essential role in the subsequent chapters. But one could cover
only the first and third sections of this chapter and then go on to Chapter
11 or 12 to study typical systems of differential equations found in appli-
cations. Chapter 13 is essentially a continuation of Chapter 12. Chapters
10-13 form the core for a course emphasizing basic models in mechanics.
Chapter 14 is an independent topic at a somewhat more advanced level
that only depends on Chapters 0-5. It develops some functional analysis
techniques and their application to finite element methods.
Baton Rouge, LA Susanne C. Brenner
Chicago, IL L. Ridgway Scott
20/07/2007
Preface to the Second Edition
This edition contains two new chapters. The first one is on the additive
Schwarz theory with applications to multilevel and domain decomposition
preconditioners, and the second one is an introduction toa posteriori error
estimators and adaptivity. We have also included a new section on an ex-
ample of a one-dimensional adaptive mesh, a new section on the discrete
Sobolevinequality andnewexercisesthroughout.Thelistofreferenceshas
also been expanded and updated.
We take this opportunity to extend thanks to everyone who provided
comments and suggestions about this book over the years, and to the Na-
tionalScienceFoundationforsupport.WealsowishtothankAchiDosanjh
and the production staff at Springer-Verlag for their patience and care.
Columbia, SC Susanne C. Brenner
Chicago, IL L. Ridgway Scott
20/02/2002
Preface to the First Edition
This book develops the basic mathematical theory of the finite element
method,themostwidelyusedtechniqueforengineeringdesignandanalysis.
Onepurposeofthisbookistoformalizebasictoolsthatarecommonlyused
by researchers in the field but never published. It is intended primarily for
mathematicsgraduatestudentsandmathematicallysophisticatedengineers
and scientists.
The book has been the basis for graduate-level courses at The Uni-
versity of Michigan, Penn State University and the University of Houston.
The prerequisite is only a course in real variables, and even this has not
been necessary for well-prepared engineers and scientists in many cases.
The book can be used for a course that provides an introduction to ba-
sicfunctionalanalysis,approximationtheoryandnumericalanalysis,while
building upon and applying basic techniques of real variable theory.
Chapters0through5formtheessentialmaterialforacourse.Chapter0
providesamicrocosmofwhatistofollow,developedintheone-dimensional
case.Chapters1through4providethebasictheory,andChapter5develops
basic applications of this theory. From this point, courses can bifurcate in
various directions. Chapter 6 provides an introduction to efficient iterative
solvers for the linear systems of finite element equations. While essential
from a practical point of view (our reason for placing it in a prominent
position), this could be skipped, as it is not essential for further chapters.
Similarly, Chapter 7, which derives error estimates in the maximum norm
and shows how such estimates can be applied to nonlinear problems, can
be skipped as desired.
Chapter8,however,hasanessentialroleinthefollowingchapters.But
onecouldcoveronlythefirstandthirdsectionsofthischapterandthengo
ontoChapter9inordertoseeanexampleofthemorecomplexsystemsof
differentialequationsthatarethenorminapplications.Chapter10depends
to some extent on Chapter 9, and Chapter 11 is essentially a continuation
of Chapter 10. Chapter 12 presents Banach space interpolation techniques
with applications to convergence results for finite element methods. This is
an independent topic at a somewhat more advanced level.
Tobemoreprecise,wedescribethreepossiblecoursepathsthatcanbe
xii Preface to the First Edition
chosen.Inallcases,thefirststepistofollowChapters0through5.Someone
interestedtopresentsomeofthe“hardestimates”ofthesubjectcouldthen
choose from Chapters 6 through 8, and 12. On the other hand, someone
interestedmoreinphysicalapplicationscouldselectfromSect.8.1,Sect.8.3
and Chapters 9 through 11. Someone interested primarily in algorithmic
efficiencyandcodedevelopmentissuescouldfollowChapters6,8,10and11.
The omissions from the book are so numerous that is hard to begin
to list them. We attempt to list the most glaring omissions for which there
are excellent books available to provide material.
We avoid time-dependent problems almost completely, partly because
of the existence of the book of (Thom´ee 1984). Our extensive development
of different types of elements and the corresponding approximation the-
ory is complementary to Thom´ee’s approach. Similarly, our development
of physical applications is limited primarily to linear systems in continuum
mechanics.Moresubstantialphysicalapplicationscanbefoundinthebook
by (Johnson 1987).
Verylittleissaidhereaboutadaptivity.Thisactiveresearchareaisad-
dressedinvariousconferenceproceedings(cf.Babuˇska,Chandra&Flaherty
1983 and Babuˇska, Zienkiewicz, Gago & de A. Oliveira 1986).
We emphasize the variety of discretizations (that is, different “ele-
ments”) that can be used, and we present them (whenever possible) as
families depending on a parameter (usually the degree of approximation).
Thus, a spirit of “high-order” approximations is developed, although we
do not consider increasing the degree of approximation (as is done in the
so-called P-method and spectral element method) as the means of obtain-
ing convergence. Rather, we focus on mesh subdivision as the convergence
parameter. The recent book by (Szabo & Babuˇska 1991) may be consulted
for alternatives in this direction.
Although we provide a brief introduction to mixed methods, the im-
portance of this subject is not appropriately reflected here. However, the
recent book by (Brezzi & Fortin 1991) can be consulted for a thorough
treatment of the subject.
We draw extensively on the book of (Ciarlet 1978), both following
many of its ideas and using it as a reference for further development of
various subjects. This book has recently been updated in (Ciarlet & Lions
1991),whichalsocontainsanexcellentsurveyofmixedmethods.Moreover,
the Handbook series to which the latter reference belongs can be expected
to provide valuable reference material in the future.
We take this opportunity to thank the many people who have helped
at various stages, and in many different ways, in the preparation of this
book. The many students who struggled through early drafts of the book
made invaluable contributions. Many readers of the preliminary versions
will find their specific suggestions incorporated.
This book was processed by the authors using the TEX macro package
from Springer-Verlag.