Table Of Content

Texts in Applied Mathematics 73 Alexandre Ern Jean-Luc Guermond Finite Elements II Galerkin Approximation, Elliptic and Mixed PDEs Texts in Applied Mathematics Volume 73 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 II Galerkin Approximation, Elliptic and Mixed PDEs 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 ISBN978-3-030-56922-8 ISBN978-3-030-56923-5 (eBook) https://doi.org/10.1007/978-3-030-56923-5 ©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 Contents Part V. Weak formulations and well-posedness 24 Weak formulation of model problems . . . . . . . . . . . . . . . . . . . . . . 3 24.1 A second-order PDE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 24.2 A first-order PDE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 24.3 A complex-valued model problem . . . . . . . . . . . . . . . . . . . . 10 24.4 Toward an abstract model problem . . . . . . . . . . . . . . . . . . . 11 25 Main results on well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 25.1 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 25.2 Lax–Milgram lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 25.3 Banach–Nečas–Babuška (BNB) theorem . . . . . . . . . . . . . . . 17 25.4 Two examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Part VI. Galerkin approximation 26 Basic error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 26.1 The Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 26.2 Discrete well-posedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 26.3 Basic error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 27 Error analysis with variational crimes . . . . . . . . . . . . . . . . . . . . . . 41 27.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 27.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 27.3 Two simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 27.4 Strang’s lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 v vi Contents 28 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 28.1 Stiffness and mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . 55 28.2 Bounds on the stiffness and mass matrices. . . . . . . . . . . . . . 58 28.3 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 29 Sparse matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 29.1 Origin of sparsity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 29.2 Storage and assembling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 29.3 Reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 30 Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 30.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 30.2 Quadrature error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 30.3 Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Part VII. Elliptic PDEs: conforming approximation 31 Scalar second-order elliptic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 97 31.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 31.2 Dirichlet boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 100 31.3 Robin/Neumann conditions . . . . . . . . . . . . . . . . . . . . . . . . . 103 31.4 Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 32 H1-conforming approximation (I). . . . . . . . . . . . . . . . . . . . . . . . . . 115 32.1 Continuous and discrete problems . . . . . . . . . . . . . . . . . . . . 115 32.2 Error analysis and best approximation in H1. . . . . . . . . . . . 117 32.3 L2-error analysis: the duality argument . . . . . . . . . . . . . . . . 120 32.4 Elliptic projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 33 H1-conforming approximation (II). . . . . . . . . . . . . . . . . . . . . . . . . 125 33.1 Non-homogeneous Dirichlet conditions. . . . . . . . . . . . . . . . . 125 33.2 Discrete maximum principle. . . . . . . . . . . . . . . . . . . . . . . . . 131 33.3 Discrete problem with quadratures . . . . . . . . . . . . . . . . . . . 135 34 A posteriori error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 34.1 The residual and its dual norm . . . . . . . . . . . . . . . . . . . . . . 141 34.2 Global upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 34.3 Local lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 34.4 Adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 35 The Helmholtz problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 35.1 Robin boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 157 35.2 Mixed boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 164 Contents vii 35.3 Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 166 35.4 H1-conforming approximation . . . . . . . . . . . . . . . . . . . . . . . 167 Part VIII. Elliptic PDEs: nonconforming approximation 36 Crouzeix–Raviart approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 175 36.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 36.2 Crouzeix–Raviart discretization . . . . . . . . . . . . . . . . . . . . . . 176 36.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 37 Nitsche’s boundary penalty method. . . . . . . . . . . . . . . . . . . . . . . . 191 37.1 Main ideas and discrete problem . . . . . . . . . . . . . . . . . . . . . 191 37.2 Stability and well-posedness. . . . . . . . . . . . . . . . . . . . . . . . . 193 37.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 38 Discontinuous Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 38.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 38.2 Symmetric interior penalty. . . . . . . . . . . . . . . . . . . . . . . . . . 200 38.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 38.4 Discrete gradient and fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 207 39 Hybrid high-order method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 39.1 Local operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 39.2 Discrete problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 39.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 40 Contrasted diffusivity (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 40.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 40.2 Discrete setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 40.3 The bilinear form n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 41 Contrasted diffusivity (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 41.1 Continuous and discrete settings . . . . . . . . . . . . . . . . . . . . . 239 41.2 Crouzeix–Raviart approximation . . . . . . . . . . . . . . . . . . . . . 241 41.3 Nitsche’s boundary penalty method. . . . . . . . . . . . . . . . . . . 243 41.4 Discontinuous Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 41.5 The hybrid high-order method . . . . . . . . . . . . . . . . . . . . . . . 247 Part IX. Vector-valued elliptic PDEs 42 Linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 42.1 Continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 42.2 Weak formulation and well-posedness . . . . . . . . . . . . . . . . . 255 42.3 H1-conforming approximation . . . . . . . . . . . . . . . . . . . . . . . 259 42.4 Further topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 viii Contents 43 Maxwell’s equations: H(curl)-approximation. . . . . . . . . . . . . . . . . 269 43.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 43.2 Weak formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 43.3 Approximation using edge elements. . . . . . . . . . . . . . . . . . . 275 44 Maxwell’s equations: control on the divergence . . . . . . . . . . . . . . 279 44.1 Functional setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 44.2 Coercivity revisited for edge elements . . . . . . . . . . . . . . . . . 282 44.3 The duality argument for edge elements . . . . . . . . . . . . . . . 286 45 Maxwell’s equations: further topics . . . . . . . . . . . . . . . . . . . . . . . . 291 45.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 45.2 Boundary penalty method in H(curl). . . . . . . . . . . . . . . . . . 292 45.3 Boundary penalty method in H1 . . . . . . . . . . . . . . . . . . . . . 298 45.4 H1-approximation with divergence control. . . . . . . . . . . . . . 299 Part X. Eigenvalue problems 46 Symmetric elliptic eigenvalue problems. . . . . . . . . . . . . . . . . . . . . 305 46.1 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 46.2 Introductory examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 47 Symmetric operators, conforming approximation . . . . . . . . . . . . . 319 47.1 Symmetric and coercive eigenvalue problems. . . . . . . . . . . . 319 47.2 H1-conforming approximation . . . . . . . . . . . . . . . . . . . . . . . 323 48 Nonsymmetric problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 48.1 Abstract theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 48.2 Conforming approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 336 48.3 Nonconforming approximation . . . . . . . . . . . . . . . . . . . . . . . 340 Part XI. PDEs in mixed form 49 Well-posedness for PDEs in mixed form . . . . . . . . . . . . . . . . . . . . 347 49.1 Model problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 49.2 Well-posedness in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 350 49.3 Saddle point problems in Hilbert spaces. . . . . . . . . . . . . . . . 354 49.4 Babuška–Brezzi theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 50 Mixed finite element approximation . . . . . . . . . . . . . . . . . . . . . . . 363 50.1 Conforming Galerkin approximation . . . . . . . . . . . . . . . . . . 363 50.2 Algebraic viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 50.3 Iterative solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Contents ix 51 Darcy’s equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 51.1 Weak mixed formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 51.2 Primal, dual, and dual mixed formulations . . . . . . . . . . . . . 385 51.3 Approximation of the mixed formulation. . . . . . . . . . . . . . . 386 52 Potential and flux recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 52.1 Hybridization of mixed finite elements. . . . . . . . . . . . . . . . . 393 52.2 Flux recovery for H1-conforming elements. . . . . . . . . . . . . . 398 53 Stokes equations: Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 53.1 Incompressible fluid mechanics. . . . . . . . . . . . . . . . . . . . . . . 405 53.2 Weak formulation and well-posedness . . . . . . . . . . . . . . . . . 407 53.3 Conforming approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 412 53.4 Classical examples of unstable pairs. . . . . . . . . . . . . . . . . . . 416 54 Stokes equations: Stable pairs (I). . . . . . . . . . . . . . . . . . . . . . . . . . 421 54.1 Proving the inf-sup condition. . . . . . . . . . . . . . . . . . . . . . . . 421 54.2 Mini element: the ðPP -bubble, P Þ pair. . . . . . . . . . . . . . . . . 424 1 1 54.3 Taylor–Hood element: the ðPP ;P Þ pair . . . . . . . . . . . . . . . . 427 2 1 54.4 Generalizations of the Taylor–Hood element . . . . . . . . . . . . 429 55 Stokes equations: Stable pairs (II) . . . . . . . . . . . . . . . . . . . . . . . . . 433 55.1 Macroelement techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 55.2 Discontinuous pressures and bubbles . . . . . . . . . . . . . . . . . . 437 55.3 Scott–Vogelius elements and generalizations . . . . . . . . . . . . 440 55.4 Nonconforming and hybrid methods . . . . . . . . . . . . . . . . . . 443 55.5 Stable pairs with QQk-based velocities . . . . . . . . . . . . . . . . . . 446 Appendix C Bijective operators in Banach spaces. .... .... .... .... ..... .... 451 C.1 Injection, surjection, bijection.... .... .... .... ..... .... 451 C.2 Banach spaces.... ..... .... .... .... .... .... ..... .... 452 C.3 Hilbert spaces.... ..... .... .... .... .... .... ..... .... 453 C.4 Duality, reflexivity, and adjoint operators.. .... ..... .... 455 C.5 Open mapping and closed range theorems.. .... ..... .... 458 C.6 Characterization of surjectivity... .... .... .... ..... .... 460 C.7 Characterization of bijectivity.... .... .... .... ..... .... 465 C.8 Coercive operators ..... .... .... .... .... .... ..... .... 467 References.... .... .... .... ..... .... .... .... .... .... ..... .... 471 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 489