Accuracy of Mathematical Models: Dimension Reduction, Homogenization, and Simplification

335 Pages·2020·4.436 MB·English

by  Sergey R. Repin and Stefan Sauter

Tracts in Mathematics 33 SS Tracts in Mathematics 33
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Sergey I. Repin
Sergey I. Repin
Stefan A. Sauter
Stefan A. Sauter
Accuracy of Mathematical Models
The expansion of scientific knowledge and the development of technology A Accuracy of
are strongly connected with quantitative analysis of mathematical mo dels. c
Accuracy and reliability are the key properties we wish to understand and c
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control.
r Mathematical Models
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This book presents a unified approach to the analysis of accuracy of c
deterministic mathematical models described by variational problems and y
partial differential equations of elliptic type. It is based on new mathematical
o
methods developed to estimate the distance between a solution of a
f Dimension Reduction,
boundary value problem and any function in the admissible functiona l class
M
associated with the problem in question. The theory is presented fo r a
wide class of elliptic variational problems. It is applied to the investigation a Homogenization, and Simplification
of modelling errors arising in dimension reduction, homogenization, t
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simplification, and various conversion methods (penalization, linearization,
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regularization, etc.). A collection of examples illustrates the performance of
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error estimates.
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ISBN 978-3-03719-206-1
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Sergey I. Repin
Stefan A. Sauter
Accuracy of
Mathematical Models
Dimension Reduction,
Homogenization, and Simplification
Authors:
Sergey I. Repin Stefan A. Sauter
Steklov Institute of Mathematics Institut für Mathematik
Russian Acadademy of Sciences Universität Zürich
Fontanka, 27 Winterthurerstr. 190
191023 St. Petersburg 8057 Zürich
Russia Switzerland
[email protected] [email protected]
2010 Mathematical Subject Classification (primary; secondary): 35-02; 35J20, 35J50, 35J60, 35J88,
49M29, 65N15, 65N85, 74K20
Key words: Modelling error, a posteriori error majorant, model simplification, dimension reduction,
homogenization, conversion of models
ISBN 978-3-03719-206-1
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and
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To our parents

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basicnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Domainsandoperators . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Spacesoffunctions . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Convexfunctionals . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Functionalinequalities . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Ho¨ldertypeinequalities . . . . . . . . . . . . . . . . . . . . 7
1.2.2 FriedrichsandPoincare´ inequalities . . . . . . . . . . . . . . 7
1.2.3 Inequalitiesforfunctionswithzeromeantraces
ontheboundary . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Korn’sinequalities . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5 Inf–Supcondition . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Computableboundsofconstants
infunctionalinequalities . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 ConstantintheFriedrichsinequality . . . . . . . . . . . . . . 16
1.3.2 ConstantsinPoincare´-typeinequalities . . . . . . . . . . . . 17
1.3.3 Constantsintrace-typeinequalities. . . . . . . . . . . . . . . 20
1.3.4 Estimatesofconstantsbasedondomaindecomposition . . . . 20
2 Distancetoexactsolutions. . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Aclassofboundaryvalueproblems . . . . . . . . . . . . . . . . . . 25
2.2 Themainerroridentity . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Errormeasure . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Decompositionoftheerrormeasure . . . . . . . . . . . . . . 31
2.2.3 ProblemswithlinearF . . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Erroridentitiesinvectorform . . . . . . . . . . . . . . . . . 42
2.2.5 Differencebetweentheexactsolutionsoftwoproblems. . . . 43
2.3 Linearproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3.1 Errorrelationsinthegeneralform . . . . . . . . . . . . . . . 44
2.3.2 Specialcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.3 Primal-dualnormsoferrorsinV (cid:2)Y(cid:2) . . . . . . . . . . . . 50
2.3.4 Errorsinthefullprimal-dualnorm . . . . . . . . . . . . . . . 52
2.3.5 Majorantasasourceofnewmodels . . . . . . . . . . . . . . 54
2.3.6 Non-homogeneousboundaryconditions . . . . . . . . . . . . 55
2.4 Applicationstoparticularmathematicalmodels. . . . . . . . . . . . 57
2.4.1 Diffusiontypemodels . . . . . . . . . . . . . . . . . . . . . 57
2.4.2 Mixedboundaryconditions . . . . . . . . . . . . . . . . . . . 59
2.4.3 Problemswithperiodicboundaryconditions . . . . . . . . . . 60
2.4.4 Advancedestimatesbasedondomaindecomposition . . . . . 61
viii Contents
2.4.5 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.6 Variationalfunctionalswithpowergrowth . . . . . . . . . . . 70
2.4.7 Stokesproblem . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.4.8 Binghamproblem . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4.9 Anothererrorestimationmethod . . . . . . . . . . . . . . . . 81
2.5 Validationofmathematicalmodels. . . . . . . . . . . . . . . . . . . 86
2.6 Errorsofnumericalapproximations . . . . . . . . . . . . . . . . . . 89
2.6.1 Two-sidedestimatesofapproximationerrors . . . . . . . . . 90
2.6.2 ReductionofthesetQ(cid:2) . . . . . . . . . . . . . . . . . . . 91
ƒ(cid:2)
2.6.3 TransformationofhR.y(cid:2)/;e i . . . . . . . . . . . . . . . . . 92
h h
2.6.4 Usingextraregularityoftheexactsolution . . . . . . . . . . 93
2.6.5 Usinganauxiliaryfinite-dimensionalproblem . . . . . . . . . 94
2.6.6 Applicationstoleastsquarestypemethods. . . . . . . . . . . 99
2.6.7 Nonconformingapproximations . . . . . . . . . . . . . . . . 101
3 Dimensionreductionmodels . . . . . . . . . . . . . . . . . . . . . . . . 103
3.1 Dimensionreduction . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Second-orderellipticproblems . . . . . . . . . . . . . . . . . . . . . 107
3.2.1 Basicproblem . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.2.2 Reducedproblem . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2.3 Errorgeneratedbydimensionreduction . . . . . . . . . . . . 110
3.2.4 Particularcases . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.3 Dimensionreductioninlinearelasticity . . . . . . . . . . . . . . . . 123
3.3.1 Theplanestressproblem . . . . . . . . . . . . . . . . . . . . 123
3.3.2 Thefunction(cid:2) . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.3.3 Behaviorofthemodellingerrorast!0 . . . . . . . . . . . 132
3.3.4 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.4 Bendingofelasticplates . . . . . . . . . . . . . . . . . . . . . . . . 136
3.4.1 Statementoftheproblem . . . . . . . . . . . . . . . . . . . . 136
3.4.2 TheKirchhoff–Loveplatemodel . . . . . . . . . . . . . . . . 137
3.4.3 Reconstructionof3Ddisplacements . . . . . . . . . . . . . . 140
3.4.4 Reconstructionof3Dstresses . . . . . . . . . . . . . . . . . 140
3.4.5 Errorestimatesforplate-typedomains . . . . . . . . . . . . . 141
3.4.6 AccuracyoftheKLplatemodel . . . . . . . . . . . . . . . . 149
3.4.7 Estimatesofthemodellingerror . . . . . . . . . . . . . . . . 150
3.4.8 Asymptoticbehaviouroftheerrormajorant . . . . . . . . . . 152
4 Modelsimplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.1 Modelsimplificationbasedontheconceptofenergy . . . . . . . . . 159
4.2 Simplificationofcoefficients . . . . . . . . . . . . . . . . . . . . . . 164
4.2.1 Second-orderellipticproblems . . . . . . . . . . . . . . . . . 165
4.2.2 Generalellipticproblem . . . . . . . . . . . . . . . . . . . . 168
4.2.3 UsingextraregularityofuQ . . . . . . . . . . . . . . . . . . . 170
Contents ix
4.2.4 AsymptoticrateofconvergenceoftheerrorestimatorE.vO/
intermsofthemeasureofthenon-resolvedgeometry . . . . . 172
4.3 Geometricalsimplification . . . . . . . . . . . . . . . . . . . . . . . 173
4.3.1 SimplificationoftheDirichletboundary . . . . . . . . . . . . 173
4.3.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.3.3 SimplificationoftheNeumannboundary . . . . . . . . . . . 179
4.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.4.1 Problemswith“rough”coefficients . . . . . . . . . . . . . . 182
4.4.2 Modelling-discretizationadaptationstrategies . . . . . . . . . 184
4.4.3 Problemswithuncertaindata . . . . . . . . . . . . . . . . . . 185
5 Elliptichomogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.2 Mathematicalhomogenizationviaasymptoticexpansions. . . . . . . 190
5.3 Propertiesofthehomogenizedproblem . . . . . . . . . . . . . . . . 195
5.3.1 Well-posednessofthehomogenizedequation . . . . . . . . . 196
5.3.2 Regularityestimatesforthehomogenizedequation . . . . . . 201
5.3.3 Regularityestimatesforthecellproblem. . . . . . . . . . . . 209
5.3.4 Convergenceofthefirst-orderapproximation . . . . . . . . . 213
5.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
5.5 Errorestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.5.1 Generalcomments . . . . . . . . . . . . . . . . . . . . . . . 221
5.5.2 Estimatesofthemodellingerror . . . . . . . . . . . . . . . . 225
5.5.3 Errorofthefullydiscretefirst-orderapproximation . . . . . . 230
5.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.6.1 Regularityandembeddingconstants . . . . . . . . . . . . . . 245
5.6.2 Modeling-discretizationstrategies . . . . . . . . . . . . . . . 247
5.6.3 Multiscaleproblems . . . . . . . . . . . . . . . . . . . . . . 248
6 Conversionofmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.1 Regularizationofmodels . . . . . . . . . . . . . . . . . . . . . . . . 251
6.1.1 Addingaregularizingterm . . . . . . . . . . . . . . . . . . . 251
6.1.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.1.3 Prox-typeregularization . . . . . . . . . . . . . . . . . . . . 259
6.2 Errorsofpenalty-typemodels . . . . . . . . . . . . . . . . . . . . . 262
6.2.1 Generalapproach . . . . . . . . . . . . . . . . . . . . . . . . 262
6.2.2 Variationalproblemsdefinedinsubspaces . . . . . . . . . . . 265
6.3 Fictitiousdomainmethods . . . . . . . . . . . . . . . . . . . . . . . 269
6.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.5 Errorsoftime-incrementalmodels . . . . . . . . . . . . . . . . . . . 278