Radon Series on Computational and Applied Mathematics 4 Managing Editor Heinz W.Engl (Linz/Vienna) Editors Hansjörg Albrecher (Linz) Ronald H.W.Hoppe (Augsburg/Houston) Karl Kunisch (Graz) Ulrich Langer (Linz) Harald Niederreiter (Singapore) Christian Schmeiser (Linz/Vienna) Sergey Repin A Posteriori Estimates for Partial Differential Equations ≥ Walter de Gruyter · Berlin · New York Author Prof.SergeyI.Repin V.A.SteklovInstituteofMathematicsatSt.Petersburg Fontanka27 191023St.Petersburg Russia E-mail:[email protected] Keywords Partialdifferentialequations,aposteriorierrorestimates,Poisson’sequation,diffusionproblems, elasticity,incompressibleviscousfluids,nonlinearproblems MathematicsSubjectClassification2000 00-02,35-02,35J05,35K05,65M15,65N15,74Bxx,76Dxx (cid:2)(cid:2) Printedonacid-freepaperwhichfallswithintheguidelines oftheANSItoensurepermanenceanddurability. ISBN 978-3-11-019153-0 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de. (cid:2)Copyright2008byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, includingphotocopy,recording,oranyinformationstorageorretrievalsystem,withoutpermission inwritingfromthepublisher. PrintedinGermany Coverdesign:MartinZech,Bremen. Typesetusingtheauthor’sLATXfiles:KayDimler,Müncheberg. E Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen. Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Aprioriandaposteriorimethodsoferrorestimation . . . . . . . . . 1 1.2 Bookstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Theerrorcontrolproblem. . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Mathematicalbackgroundandnotation. . . . . . . . . . . . . . . . . 8 1.4.1 Vectorsandtensors . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Spacesoffunctions . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.4 Convexfunctionals . . . . . . . . . . . . . . . . . . . . . . . 17 2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1 ErrorindicatorbyRunge . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Prager–Syngeestimate . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Mikhlinestimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Ostrowskiestimatesforcontractivemappings . . . . . . . . . . . . . 26 2.5 Errorestimatesbasedonmonotonicity . . . . . . . . . . . . . . . . . 30 2.6 Aposteriorierrorindicatorsforfiniteelementapproximations . . . . 31 2.6.1 Explicitresidualmethods. . . . . . . . . . . . . . . . . . . . 32 2.6.2 Implicitresidualmethods. . . . . . . . . . . . . . . . . . . . 35 2.6.3 A posteriori estimates based on post-processing of approxi- matesolutions . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.4 Aposteriorimethodsusingadjointproblems . . . . . . . . . 42 3 Poisson’sequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Thevariationalmethod . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Themethodofintegralidentities . . . . . . . . . . . . . . . . . . . . 50 3.3 Propertiesofaposterioriestimates . . . . . . . . . . . . . . . . . . . 52 3.4 Two-sidedboundsincombinednorms . . . . . . . . . . . . . . . . . 57 3.5 Modificationsofestimates . . . . . . . . . . . . . . . . . . . . . . . 59 3.5.1 Galerkinapproximations . . . . . . . . . . . . . . . . . . . . 59 3.5.2 Advancedformsoferrorbounds . . . . . . . . . . . . . . . . 60 3.5.3 Decompositionofthedomain . . . . . . . . . . . . . . . . . 62 3.5.4 Estimateswithpartiallyequilibratedfluxes . . . . . . . . . . 64 3.6 Howcanoneusefunctionalaposterioriestimatesinpracticalcompu- tations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.6.1 Post-processingoffluxes . . . . . . . . . . . . . . . . . . . . 65 vi Contents 3.6.2 Rungetypeestimate . . . . . . . . . . . . . . . . . . . . . . 66 3.6.3 Minimizationofthemajorant . . . . . . . . . . . . . . . . . 66 3.6.4 Errorindicatorsgeneratedbyerrormajorants . . . . . . . . . 70 4 Linearellipticproblems . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Two-sidedestimatesforstationarydiffusionproblem . . . . . . . . . 75 4.1.1 Estimatesforproblemswithmixedboundaryconditions . . . 75 4.1.2 Modificationsofestimates . . . . . . . . . . . . . . . . . . . 78 4.1.3 EstimatesforproblemswithNeumannboundarycondition . . 80 4.2 Thestationaryreaction-diffusionproblem . . . . . . . . . . . . . . . 81 4.3 Diffusionproblemswithconvectiveterm . . . . . . . . . . . . . . . . 87 4.3.1 Thestationaryconvection-diffusionproblem . . . . . . . . . 88 4.3.2 Thereaction-convection-diffusionproblem . . . . . . . . . . 92 4.3.3 Specialcasesandmodifications . . . . . . . . . . . . . . . . 95 4.3.4 Estimatesforfluxes . . . . . . . . . . . . . . . . . . . . . . . 99 4.4 Notesforthechapter . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1 Thelinearelasticityproblem . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Estimatesfordisplacements . . . . . . . . . . . . . . . . . . . . . . 107 5.3 Estimatesforstresses . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4 Isotropiclinearelasticity . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.1 3Dproblems . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.4.2 Theplanestressproblem . . . . . . . . . . . . . . . . . . . . 111 5.4.3 Theplanestrainproblem . . . . . . . . . . . . . . . . . . . . 113 5.4.4 Erroroftheplanestressmodel . . . . . . . . . . . . . . . . . 114 5.5 Notesforthechapter . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6 Incompressibleviscousfluids . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1 TheStokesproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 AposterioriestimatesforthestationaryStokesproblem . . . . . . . . 123 6.2.1 Estimatesforthevelocityfield . . . . . . . . . . . . . . . . . 123 6.2.2 Estimatesforpressure . . . . . . . . . . . . . . . . . . . . . 127 6.2.3 Estimatesforstresses . . . . . . . . . . . . . . . . . . . . . . 128 6.2.4 Estimatesincombinednorms . . . . . . . . . . . . . . . . . 128 6.2.5 Lowerboundsoferrors . . . . . . . . . . . . . . . . . . . . . 130 6.2.6 Mixedboundaryconditions . . . . . . . . . . . . . . . . . . 131 6.2.7 Problemsforalmostincompressiblefluids . . . . . . . . . . . 137 6.2.8 ProblemswiththeconditiondivuD(cid:2) . . . . . . . . . . . . 139 6.3 GeneralizedStokesproblem . . . . . . . . . . . . . . . . . . . . . . 140 6.3.1 Estimatesforsolenoidalapproximations . . . . . . . . . . . . 141 6.3.2 Estimatesfornonsolenoidalfields . . . . . . . . . . . . . . . 145 6.3.3 Estimatesforthepressurefield . . . . . . . . . . . . . . . . . 146 Contents vii 6.3.4 Errorminorant . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3.5 Modelswithpolymerization . . . . . . . . . . . . . . . . . . 148 6.3.6 Modelswithrotation . . . . . . . . . . . . . . . . . . . . . . 149 6.4 TheOseenproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.5 StationaryNavier–Stokesproblemford D2 . . . . . . . . . . . . . 153 6.6 Notesforthechapter . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.1 Linearellipticproblem . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.1.1 Thevariationalmethod . . . . . . . . . . . . . . . . . . . . . 159 7.1.2 Themethodofintegralidentities . . . . . . . . . . . . . . . . 164 7.1.3 Errorestimatesforthedualvariable . . . . . . . . . . . . . . 168 7.1.4 Two-sidedestimatesforcombinednorms . . . . . . . . . . . 168 7.2 Ellipticproblemswithlowerterms . . . . . . . . . . . . . . . . . . . 171 7.3 Problemswithsolutionsdefinedinsubspaces . . . . . . . . . . . . . 173 7.3.1 Abstractproblem . . . . . . . . . . . . . . . . . . . . . . . . 173 7.3.2 Estimateforapproximationslyinginthesubspace . . . . . . 173 7.3.3 Estimateforapproximationslyingintheenergyspace . . . . 174 7.4 Derivationofaposterioriestimatesfromsaddlepointrelations . . . . 176 8 Nonlinearproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.1 Variationalinequalities . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.1.1 Variationalinequalitiesofthefirstkind . . . . . . . . . . . . 179 8.1.2 Variationalinequalitiesofthesecondkind . . . . . . . . . . . 185 8.2 Generalellipticproblem. Variationalmethod. . . . . . . . . . . . . . 186 8.3 Generalellipticproblem. Nonvariationalmethod . . . . . . . . . . . 191 8.4 Aposterioriestimatesforspecialclassesofnonlinearellipticproblems 196 8.4.1 ˛-Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.4.2 Problemswithnonlinearboundaryconditions . . . . . . . . . 201 8.4.3 GeneralizedNewtonianfluids . . . . . . . . . . . . . . . . . 211 8.5 Notesforthechapter . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9 Otherproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9.1 Differentialequationsofhigherorder . . . . . . . . . . . . . . . . . 218 9.2 Equationswiththeoperatorcurl . . . . . . . . . . . . . . . . . . . . 224 9.3 Evolutionaryproblems . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.3.1 Thelinearevolutionaryproblem . . . . . . . . . . . . . . . . 229 9.3.2 Firstformoftheerrormajorant . . . . . . . . . . . . . . . . 231 9.3.3 Secondformoftheerrormajorant . . . . . . . . . . . . . . . 235 9.3.4 Equivalenceofthedeviationandmajorant . . . . . . . . . . . 238 9.3.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.4 Aposterioriestimatesforoptimalcontrolproblems . . . . . . . . . . 242 9.4.1 Two-sidedboundsforcostfunctionals . . . . . . . . . . . . . 243 viii Contents 9.4.2 Estimatesforstateandcontrolfunctions . . . . . . . . . . . . 248 9.4.3 Estimateinacombinednorm . . . . . . . . . . . . . . . . . 251 9.4.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 252 9.4.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.5 Estimatesfornonconformingapproximations . . . . . . . . . . . . . 254 9.5.1 Estimatesbasedonprojectingtotheenergyspace . . . . . . . 255 9.5.2 EstimatesbasedontheHelmholtzdecomposition . . . . . . . 257 9.5.3 AccuracyofapproximationsobtainedbytheTrefftzmethod . 263 9.5.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.6 Uncertaindata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.6.2 Errorscausedbyindeterminacyincoefficients . . . . . . . . 270 9.6.3 Errorsowingtouncertain(cid:3) . . . . . . . . . . . . . . . . . . 276 9.6.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.7 Errorestimatesintermsoffunctionalsandnonenergynorms . . . . . 279 9.7.1 Generalframework . . . . . . . . . . . . . . . . . . . . . . . 279 9.7.2 Estimatesinlocalnorms . . . . . . . . . . . . . . . . . . . . 280 9.7.3 Estimatesintermsoflinearfunctionals . . . . . . . . . . . . 281 9.7.4 EstimatesbasedonthePoincare´ inequality . . . . . . . . . . 284 9.7.5 Estimatesbasedonmultiplicativeinequalities . . . . . . . . . 285 9.7.6 Estimatesbasedonthemaximumprinciple . . . . . . . . . . 285 9.7.7 Estimatesinweightednorms . . . . . . . . . . . . . . . . . . 287 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Preface Puremathematicianssometimesaresatisfiedwithshow- ingthatthenon-existenceofasolutionimpliesalogical contradiction,whileengineersmightconsideranumer- icalresultastheonlyreasonablegoal. Suchonesided viewsseemtoreflecthumanlimitationsratherthanob- jectivevalues.Initselfmathematicsisanindivisibleor- ganismunitingtheoreticalcontemplationandactiveap- plication. R. COURANT[112] Partialdifferentialequations(PDE’s)wereintroducedasmathematicalmodelsofvar- iousphysicalphenomena. Inthe20thcentury,thetheoryofdifferentialequationswas mainly developed in the context of an a priori conception that can be expressed by thetriad: existence, regularity, andapproximation. Init, theaccentismadeonapri- orimathematicalanalysis,andnumericalexperimentisoftenregardedastheverylast (andinasensetechnical)step,whichismorerelatedtopracticalapplicationsthanto theory. Acertainrevisionofviewshasstarted20–30yearsago. Itwasstimulatedbyrapid development of numerical methods for PDE’s. The experience accumulated in this area shows that a priori methods provide only one part of the information necessary foracomprehensiveanalysisofmodelsbasedonPDE’s. Ifdifferentialequationsare considerednotasaself-containedbranchofpuremathematicsbutastoolsconsigned to serve natural sciences, then the imperfection of purely a priori analysis is easy to observe. For example, almost all results of regularity theory and asymptotic analysis have a qualitative meaning and are addressed to the whole class (or a subclass) of boundary value problems considered. However, in the numerical experiment we al- ways deal with an approximate solution of a particular problem the quality of which must be certified by a certain quantitative criterion. The latter task calls for further development of different mathematical methods focused on a posteriori analysis of approximatesolutions. The need for new mathematical approaches to the analysis of PDE’s is motivated notonlyby“routine”arguments(suchasgettingaccuratenumericalapproximations). Thereisamorefundamentalproblem: validationofmathematicalmodels. Certainly, itcanbesolvedonlybyjointeffortsofmathematiciansandspecialistsinaparticular x Preface naturalscienceandaposteriorierrorcontrolmethodsabletoguaranteethereliability ofmathematicalexperimentsmustplayanessentialroleinsuchresearch. This book can be viewed as an introduction to a posteriori error estimation theory for PDE’s, which is now in the process of formation and development. It includes an extended version of the lecture course “A posteriori estimates and adaptivity in continuum mechanics” that was prepared for the Special Radon Semester organized in 2005 by the Radon Institute of Computational and Applied Mathematics in Linz. That course was based on earlier lectures delivered in 2000–2001 for students of the St.PetersburgPolytechnicalUniversity(Russia)andin2003forstudentsandscientific researchers of the University of Jyva¨skyla¨ (Finland) and the University of Houston (USA).In2006, IreadamodifiedversionofthecourseattheHelsinkiUniversityof Technology (Finland) and in 2007 at the University of Valenciennes (France). The work with lectures was also supported by the DAAD program of Germany and FIM (Switzerland)duringlong-termvisitstotheUniversityofSaarbru¨cken(Germany)and theSwissFederalInstituteofTecnology(ETH,Zurich). For these years, the content and structure of the text varied. However, the main line of it remains the same: for each class of boundary value problems, a posteriori estimates are derived by purely functional methods, which are used in the theory of PDE’s for analysis of the corresponding differential equations. In other words, the methodsuggestedforderivingaposteriorierrorestimates(aswellasmethods,which study existence and regularity) exploits specific featuresof a particular mathematical problem,butdoesnotattractpropertiesofapproximatesolutions,mesh,andnumerical methodused(thelatterinformationcanbeutilizedlater). Asaresult,weobtainesti- mates that contain no mesh-dependent constants and are valid for any approximation fromthecorrespondingenergyspace. Thisnewfunctionalapproachtotheaposteriori errorestimationdevelopedinthelastdecadeisthemainsubjectofthebook. A posteriori estimates of the functional type came about from two sources. The variational statement of the problem (if it has variational form) generates the first derivationmethod,whichiswellexposedinmanypapersandinthebook[244]. The second method is based on transformations of integral identities that define general- izedsolutions. MostresultsexposedinChapters3–9areobtainedwiththehelpofthis “nonvariational”method. Themainideaofitisbrieflyasfollows: An integral identity (variational inequality) that defines a generalized solution is a source of guaranteed and computable bounds of the difference between this solution andanyfunctionfromthecorrespondingenergyspace. Chapter2containsaconciseoverviewofvariousaposteriorierrorestimationmeth- ods developed in the 20th century, which are different from those considered in sub- sequent chapters. Its purpose is to give only a general presentation and to discuss severalapproachestotheerrorcontrolproblem. Thereaderinterestedintheirdetailed investigationisreferredtorelevantliterature. In Chapter 3, the basic ideas of the functional approach are explained with the paradigm of a simple elliptic problem. The next chapters are devoted to particular