Table Of ContentRadon 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
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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
