Table Of ContentTexts in Applied Mathematics 72
Alexandre Ern
Jean-Luc Guermond
Finite Elements I
Approximation and Interpolation
Texts in Applied Mathematics
Volume 72
Editors-in-Chief
Anthony Bloch, University of Michigan, Ann Arbor, MI, USA
Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA
Alain Goriely, University of Oxford, Oxford, UK
Leslie Greengard, New York University, New York, NY, USA
Series Editors
J. Bell, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
R. Kohn, New York University, New York, NY, USA
P. Newton, University of Southern California, Los Angeles, CA, USA
C. Peskin, New York University, New York, NY, USA
R. Pego, Carnegie Mellon University, Pittsburgh, PA, USA
L. Ryzhik, Stanford University, Stanford, CA, USA
A. Singer, Princeton University, Princeton, NJ, USA
A Stevens, University of Münster, Münster, Germany
A. Stuart, University of Warwick, Coventry, UK
T. Witelski, Duke University, Durham, NC, USA
S. Wright, University of Wisconsin, Madison, WI, USA
The mathematization of allsciences,the fading oftraditional scientificboundaries,
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Alexandre Ern Jean-Luc Guermond
(cid:129)
Finite Elements I
Approximation and Interpolation
123
Alexandre Ern Jean-Luc Guermond
CERMICS Department ofMathematics
Ecoledes Ponts andINRIA Paris Texas A&MUniversity
Marnela ValléeandParis, France CollegeStation, TX,USA
ISSN 0939-2475 ISSN 2196-9949 (electronic)
Textsin AppliedMathematics
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Preface
Althoughtherootsofthe“FiniteElementMethod”canbefoundinthework
of Courant [84], the method really took off in the 1950’s when engineers
started to solve numerically structural mechanics problems in civil engi-
neering and in aeronautics. Since then, finite elements have become ubiqui-
tousincomputationalsciencesandengineering.Numerousacademictoolbox
and commercial codes based on the finite element method have been devel-
opedovertheyearsandarenowavailabletoalargepublic.Numerousbooks,
textbooks, and myriads of technical papers, articles, and conference
proceedings have been written on the topic.
We have contributed to this flow in 2004 by publishing Theory and
Practice of Finite Elements, in the Applied Mathematical Sciences series,
volume 159. The approach we adopted at that time was first to present the
finiteelementmethodasaninterpolationtool,thentoillustratetheideathat
finite elements can be efficiently used to approximate partial differential
equations other than the Laplace equation and in particular problems for
which the Lax–Milgram lemma is not the ultimate paradigm. One objective
of Theory and Practice of Finite Elements was to put the emphasis on the
infsup conditions developed by Babuška in 1970 in the context of finite
elementmethods[14]andstatedinatheoreticalworkbyNečasin1962[150].
These inf-sup conditions are necessary and sufficient conditions for the
well-posedness of any linear problem set in Banach spaces. From the func-
tional analysis point of view, the inf-sup conditions are a rephrasing of two
fundamental theorems by Banach: the closed range theorem and the open
mapping theorem. For this reason, we called these conditions the Banach
NečasBabuška(BNB)theorem.TheideawefollowedinTheoryandPractice
of Finite Elements was to expose fundamental concepts while staying con-
nected with practical topics such as applications to several PDEs and
implementation aspects of the finite element method.
v
vi Preface
Thepresentwork,calledFiniteElementsandorganizedinthreevolumes,
startedasasecondeditionofTheoryandPracticeofFiniteElementsatthe
invitation of Springer editors, but as we progressed in the rewriting and the
reorganizing of the material, an entirely new project emerged. We tried to
preserve the spirit of Theory and Practice of Finite Elements by covering
fundamental aspects in approximation theory and by thoroughly exploring
applications and implementation details, but Finite Elements is definitely not
a re-edition of Theory and Practice of Finite Elements. This new book is
meant to be used as a graduate textbook and as a reference for researchers
and engineers.
The bookis divided into three volumes. Volume I focuses on fundamental
ideas regarding the construction of finite elements and their approximation
properties. We have decided to start Volume I with four chapters on func-
tional analysis which we think could be useful to readers who may not be
familiar with Lebesgue integration and weak derivatives. The purpose
of these chapters is not to go through arduous technical details, but to
familiarize the reader with the functional analysis language. These four
chapters are packed with examples and counterexamples which we think
should convince the reader of the relevance of the material. Volume I also
reviews important implementation details that must be taken care of when
either developing or using a finite element toolbox, like the orientation of
meshes,andtheenumerationofthegeometricentities(vertices,edges,faces,
cells) or the enumeration of the degrees of freedom. Volume I contains two
appendices highlighting basic facts on Banach and Hilbert spaces and on
differential calculus.
Volume II starts with fundamental results on well-posed weak formula-
tions and their approximation by the Galerkin method. Key results are the
BNB Theorem, Céa’s and Strang’s lemmas(and their variants) forthe error
analysis, and the duality argument by Aubin and Nitsche. Important
implementation aspects regarding quadratures, linear algebra, and assem-
blingarealsocovered.TherestofVolumeIIfocusesonapplicationstoPDEs
where a coercivity property is available. Various conforming and noncon-
formingapproximationtechniquesareexposed(Galerkin,boundarypenalty,
Crouzeix–Raviart, discontinuous Galerkin, hybrid high-order methods). The
applicationsconsideredareellipticPDEs(diffusion,elasticity,theHelmholtz
problem, Maxwell’s equations), eigenvalue problems for elliptic PDEs, and
PDEs in mixed form (Darcy and Stokes flows). Volume II contains one
appendix collecting fundamental results on the surjectivity, bijectivity, and
coercivity of linear operators in Banach spaces.
VolumeIIIdevelopsmoreadvancedtopics.Thefirstquarterofthevolume
focuses on symmetric positive systems offirst-order PDEs called Friedrichs’
systems. Examples include advection and advection-diffusion equations and
various PDEs written in mixed form (Darcy and Stokes flows, Maxwell’s
equations). One salient aspect of this first part of the volume is the com-
prehensive and unified treatment of many stabilization techniques from the
Preface vii
literature.TheremainingofVolumeIIIdealswithtime-dependentproblems:
parabolicequations(suchastheheatequation),evolutionequationswithout
coercivity (Stokes flows, Friedrichs’ systems), and nonlinear hyperbolic
equations (scalar conservation equations, hyperbolic systems).
The book is organized into 83 chapters, most of them composed of 10–16
pages, and each chapter is accompanied by exercises. The three volumes
contain altogether over 500 exercises with all the solutions available online.
For researchers and engineers, the division in short chapters is meant to
isolate the key ideas and the most important results. The chapters are rela-
tively independent from each other and the book is not meant to be read
linearly.Eachvolumeissupplementedwithalonglistofreferences.Inorder
to help the reader, we try to pinpoint the exact chapter, section or theorem
each time we refer to a book. Short literature reviews are also included in
most of the chapters as well.
Whenused asatextbook,thedivisioninshortchapters ismeant to bean
aidtoteachersandstudents.Theobjectiveisthatonechaptercanbecovered
intimeunitsof1hto1h15.Thesalientideascanbedevelopedandexposedin
class, while the rest of the chapter can be assigned as reading material. The
exercises are important complements, and teachers are encouraged to use
some of the exercises in class. Whenever possible the exercises have been
divided into elementary steps with enough hints to be doable by reasonably
assiduous students. The book is well adapted to graduate flipped classes as
well. A significant portion of the material presented in the book has been
taught in graduate classes at Texas A&M, École nationale des ponts et
chaussées,Écolepolytechnique,andInstitutHenriPoincaré.Aboutonethird
ofthematerialhasactuallybeentaughtbythestudentsthemselvesinflipped
classes. The book can be used in many teaching contexts. Among various
possibilitiesitcanbeusedtoteachthemathematicalbasesoffiniteelements
atanintroductorylevel,itcanalsobeusedtoteachpracticalimplementation
aspects (mesh generation, enumeration, orientation, quadratures, assem-
bling), and it can be used to teach sophisticated approximation techniques
overawiderangeofproblems(ellipticPDEs,mixedPDEs,first-orderPDEs,
eigenvalue problems, parabolic PDEs, hyperbolic conservation equations).
A good part of the material is quite standard, but we have also inserted
concepts and ideas which, without being entirely new, will possibly convey
someflavorofnoveltytothereader.Forinstance,wehavedevelopedinsome
detailsandprovidedexamplesonhowtoorientmeshesandonhowtheusual
differentialoperators,aswellasnormalandtangentvectors,aretransformed
bygeometricmappings.Wehavedevelopedastep-by-stepconstructionofthe
usual conforming finite element subspaces by means of the notion of connec-
tivity classes, and we have illustrated this notion by numerous examples.
Furthermore,wehaveincludedtwochaptersonquasi-interpolationwherewe
tried to develop a fresh and unifying viewpoint on the construction of
quasi-interpolationoperatorsforallthescalar-valuedandvector-valuedfinite
elementsconsideredinthebook.Wehavealsomadeanefforttoworkasmuch
viii Preface
aspossiblewithdimensionallyconsistentexpressions.Althoughthismaylead
toslightlymorecomplexstatementsfornormsanderrorestimates,webelieve
that the present choice is important to understand the various physically
relevantregimesin themodel problems.
Someofthetechniquesthatareusedtoprovestabilityanderrorestimates,
withoutbeingentirelynew,are,inouropinion,notstandardintheliterature
at the time of this writing. In particular, the techniques that we use are
essentiallydesignedtoinvokeaslittleaprioriregularityfromthesolutionas
possible.Onesalientexampleistheanalysisofnonconformingapproximation
techniques for diffusion problems with contrasted coefficients, and another
one concerns Maxwell’s equations also in materials with contrasted proper-
ties. Moreover, we give a unified analysis of first-order PDEs by means of
Friedrichs’ systems, and we show that a large class of stabilization methods
proposed in the literature so far are more or less equivalent, whether the
approximation is continuous or discontinuous. We conclude Volume III by a
series of chapters on time-evolution problems, which give a somewhat new
perspective on the analysis of well-known time-stepping methods. The last
five chapters on hyperbolic equations, we hope, should convince the reader
that continuous finite elements are good candidates to solve this class of
problems where finite volumes have so far taken the lion’s share.
Although the reference list is quite long (about 200, 400, and 300 biblio-
graphic entries in each volume, respectively), the finite element literature is
so prolific that we have not been able to cite all the relevant contributions.
Anyway, our objective was not to be exhaustive and to write complete
reviewsofthetopicsathandbuttoisolatethekeyprinciplesandideasandto
refer the reader to the references we are the most familiar with at the time
of the writing.
Acknowledgments.Weare indebtedtomanycolleaguesandformerstudents
for valuable discussions and comments on the manuscript (W. Bangerth,
A.Bonito,E.Burman,A.Demlow,P.Minev,R.Nochetto,B.Popov,A.Till,
M. Vohralík, P. Zanotti). We are grateful to all the students who helped us
improvetheorganizationandthecontentofthebookthroughtheirfeedback.
Finally, we thank École nationale des ponts et chaussées, Institut Henri
Poincaré, the International chair program of INRIA, the Mobil Chair in
Computational Science, and the Institute of Scientific Computing at Texas
A&M University for the material and financial support provided at various
stages in this project.
Paris, France Alexandre Ern
College Station, TX, USA Jean-Luc Guermond
June 2020
Contents
Part I. Elements offunctional analysis
1 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Heuristic motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Lebesgue measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Weak derivatives and Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Differentiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Key properties: density and embedding . . . . . . . . . . . . . . . . 21
3 Traces and Poincaré inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Lipschitz sets and domains. . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Traces as functions at the boundary . . . . . . . . . . . . . . . . . . 30
3.3 Poincaré–Steklov inequalities . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Distributions and duality in Sobolev spaces. . . . . . . . . . . . . . . . . . 39
4.1 Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Negative-order Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Normal and tangential traces . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Part II. Introduction to finite elements
5 Main ideas and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Finite element as a triple . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Interpolation: finite element as a quadruple. . . . . . . . . . . . . 52
ix
