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Texts 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, theimpactofcomputertechnology,thegrowingimportanceofcomputermodelling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The aimofthisseriesistoprovidesuchtextbooksinappliedmathematicsforthestudent scientist. Books should be well illustrated and have clear exposition and sound pedagogy. Large number of examples and exercises at varying levels are recommended. TAM publishes textbooks suitable for advanced undergraduate and beginning graduate courses, and complements the Applied Mathematical Sciences (AMS)series,whichfocusesonadvancedtextbooksandresearch-levelmonographs. More information about this series at http://www.springer.com/series/1214 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 ISBN978-3-030-56340-0 ISBN978-3-030-56341-7 (eBook) https://doi.org/10.1007/978-3-030-56341-7 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 functional 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 functional 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 properties. 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