A posteriori estimates for partial differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
de Gruyter
2008
|
| Schriftenreihe: | Radon series on computational and applied mathematics
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 316 S. graph. Darst. |
| ISBN: | 9783110191530 |
Internformat
MARC
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| 100 | 1 | |a Repin, Sergej I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A posteriori estimates for partial differential equations |c Sergey Repin |
| 264 | 1 | |a Berlin [u.a.] |b de Gruyter |c 2008 | |
| 300 | |a XI, 316 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Radon series on computational and applied mathematics |v 4 | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Error analysis (Mathematics) | |
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| 830 | 0 | |a Radon series on computational and applied mathematics |v 4 |w (DE-604)BV023335470 |9 4 | |
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Datensatz im Suchindex
| _version_ | 1804137335421403136 |
|---|---|
| adam_text | Contents
Preface ix
1 Introduction 1
1.1 A priori and a posteriori methods of error estimation 1
1.2 Book structure 2
1.3 The error control problem 5
1.4 Mathematical background and notation 8
1.4.1 Vectors and tensors 8
1.4.2 Spaces of functions 11
1.4.3 Inequalities 15
1.4.4 Convex functionals 17
2 Overview 22
2.1 Error indicator by Runge 22
2.2 Prager-Synge estimate 23
2.3 Mikhlin estimate 25
2.4 Ostrowski estimates for contractive mappings 26
2.5 Error estimates based on monotonicity 30
2.6 A posteriori error indicators for finite element approximations .... 31
2.6.1 Explicit residual methods 32
2.6.2 Implicit residual methods 35
2.6.3 A posteriori estimates based on post-processing of approxi¬
mate solutions 37
2.6.4 A posteriori methods using adjoint problems 42
3 Poisson s equation 45
3.1 The variational method 45
3.2 The method of integral identities 50
3.3 Properties of a posteriori estimates 52
3.4 Two-sided bounds in combined norms 57
3.5 Modifications of estimates 59
3.5.1 Galerkin approximations 59
3.5.2 Advanced forms of error bounds 60
3.5.3 Decomposition of the domain 62
3.5.4 Estimates with partially equilibrated fluxes 64
3.6 How can one use functional a posteriori estimates in practical compu¬
tations? 65
3.6.1 Post-processing of fluxes 65
3.6.2 Runge type estimate 66
3.6.3 Minimization of the majorant 66
3.6.4 Error indicators generated by error majorants 70
4 Linear elliptic problems 75
4.1 Two-sided estimates for stationary diffusion problem 75
4.1.1 Estimates for problems with mixed boundary conditions ... 75
4.1.2 Modifications of estimates 78
4.1.3 Estimates for problems with Neumann boundary condition . . 80
4.2 The stationary reaction-diffusion problem 81
4.3 Diffusion problems with convective term 87
4.3.1 The stationary convection-diffusion problem 88
4.3.2 The reaction-convection-diffusion problem 92
4.3.3 Special cases and modifications 95
4.3.4 Estimates for fluxes 99
4.4 Notes for the chapter 103
5 Elasticity 104
5.1 The linear elasticity problem 104
5.2 Estimates for displacements 107
5.3 Estimates for stresses 109
5.4 Isotropic linear elasticity 110
5.4.1 3D problems 110
5.4.2 The plane stress problem Ill
5.4.3 The plane strain problem 113
5.4.4 Error of the plane stress model 114
5.5 Notes for the chapter 116
6 Incompressible viscous fluids 117
6.1 The Stokes problem 117
6.2 A posteriori estimates for the stationary Stokes problem 123
6.2.1 Estimates for the velocity field 123
6.2.2 Estimates for pressure 127
6.2.3 Estimates for stresses 128
6.2.4 Estimates in combined norms 128
6.2.5 Lower bounds of errors 130
6.2.6 Mixed boundary conditions 131
6.2.7 Problems for almost incompressible fluids 137
6.2.8 Problems with the condition div u = p 139
6.3 Generalized Stokes problem 140
6.3.1 Estimates for solenoidal approximations 141
6.3.2 Estimates for nonsolenoidal fields 145
6.3.3 Estimates for the pressure field 146
6.3.4 Error minorant 148
6.3.5 Models with polymerization 148
6.3.6 Models with rotation 149
6.4 The Oseen problem 151
6.5 Stationary Navier-Stokes problem for d = 2 153
6.6 Notes for the chapter 156
7 Generalizations 158
7.1 Linear elliptic problem 158
7.1.1 The variational method 159
7.1.2 The method of integral identities 164
7.1.3 Error estimates for the dual variable 168
7.1.4 Two-sided estimates for combined norms 168
7.2 Elliptic problems with lower terms 171
7.3 Problems with solutions defined in subspaces 173
7.3.1 Abstract problem 173
7.3.2 Estimate for approximations lying in the subspace 173
7.3.3 Estimate for approximations lying in the energy space .... 174
7.4 Derivation of a posteriori estimates from saddle point relations .... 176
8 Nonlinear problems 178
8.1 Variational inequalities 178
8.1.1 Variational inequalities of the first kind 179
8.1.2 Variational inequalities of the second kind 185
8.2 General elliptic problem. Variational method 186
8.3 General elliptic problem. Nonvariational method 191
8.4 A posteriori estimates for special classes of nonlinear elliptic problems 196
8.4.1 a-Laplacian 196
8.4.2 Problems with nonlinear boundary conditions 201
8.4.3 Generalized Newtonian fluids 211
8.5 Notes for the chapter 214
9 Other problems 218
9.1 Differential equations of higher order 218
9.2 Equations with the operator curl 224
9.3 Evolutionary problems 229
9.3.1 The linear evolutionary problem 229
9.3.2 First form of the error majorant 231
9.3.3 Second form of the error majorant 235
9.3.4 Equivalence of the deviation and majorant 238
9.3.5 Comments 240
9.4 A posteriori estimates for optimal control problems 242
9.4.1 Two-sided bounds for cost functionals 243
9.4.2 Estimates for state and control functions 248
9.4.3 Estimate in a combined norm 251
9.4.4 Generalizations 252
9.4.5 Comments 254
9.5 Estimates for nonconforming approximations 254
9.5.1 Estimates based on projecting to the energy space 255
9.5.2 Estimates based on the Helmholtz decomposition 257
9.5.3 Accuracy of approximations obtained by the Trefftz method . 263
9.5.4 Comments 264
9.6 Uncertain data 265
9.6.1 Introduction 265
9.6.2 Errors caused by indeterminacy in coefficients 270
9.6.3 Errors owing to uncertain Q 276
9.6.4 Comments 278
9.7 Error estimates in terms of functionals and nonenergy norms 279
9.7.1 General framework 279
9.7.2 Estimates in local norms 280
9.7.3 Estimates in terms of linear functionals 281
9.7.4 Estimates based on the Poincare inequality 284
9.7.5 Estimates based on multiplicative inequalities 285
9.7.6 Estimates based on the maximum principle 285
9.7.7 Estimates in weighted norms 287
Bibliography 291
Index 314
|
| adam_txt |
Contents
Preface ix
1 Introduction 1
1.1 A priori and a posteriori methods of error estimation 1
1.2 Book structure 2
1.3 The error control problem 5
1.4 Mathematical background and notation 8
1.4.1 Vectors and tensors 8
1.4.2 Spaces of functions 11
1.4.3 Inequalities 15
1.4.4 Convex functionals 17
2 Overview 22
2.1 Error indicator by Runge 22
2.2 Prager-Synge estimate 23
2.3 Mikhlin estimate 25
2.4 Ostrowski estimates for contractive mappings 26
2.5 Error estimates based on monotonicity 30
2.6 A posteriori error indicators for finite element approximations . 31
2.6.1 Explicit residual methods 32
2.6.2 Implicit residual methods 35
2.6.3 A posteriori estimates based on post-processing of approxi¬
mate solutions 37
2.6.4 A posteriori methods using adjoint problems 42
3 Poisson's equation 45
3.1 The variational method 45
3.2 The method of integral identities 50
3.3 Properties of a posteriori estimates 52
3.4 Two-sided bounds in combined norms 57
3.5 Modifications of estimates 59
3.5.1 Galerkin approximations 59
3.5.2 Advanced forms of error bounds 60
3.5.3 Decomposition of the domain 62
3.5.4 Estimates with partially equilibrated fluxes 64
3.6 How can one use functional a posteriori estimates in practical compu¬
tations? 65
3.6.1 Post-processing of fluxes 65
3.6.2 Runge type estimate 66
3.6.3 Minimization of the majorant 66
3.6.4 Error indicators generated by error majorants 70
4 Linear elliptic problems 75
4.1 Two-sided estimates for stationary diffusion problem 75
4.1.1 Estimates for problems with mixed boundary conditions . 75
4.1.2 Modifications of estimates 78
4.1.3 Estimates for problems with Neumann boundary condition . . 80
4.2 The stationary reaction-diffusion problem 81
4.3 Diffusion problems with convective term 87
4.3.1 The stationary convection-diffusion problem 88
4.3.2 The reaction-convection-diffusion problem 92
4.3.3 Special cases and modifications 95
4.3.4 Estimates for fluxes 99
4.4 Notes for the chapter 103
5 Elasticity 104
5.1 The linear elasticity problem 104
5.2 Estimates for displacements 107
5.3 Estimates for stresses 109
5.4 Isotropic linear elasticity 110
5.4.1 3D problems 110
5.4.2 The plane stress problem Ill
5.4.3 The plane strain problem 113
5.4.4 Error of the plane stress model 114
5.5 Notes for the chapter 116
6 Incompressible viscous fluids 117
6.1 The Stokes problem 117
6.2 A posteriori estimates for the stationary Stokes problem 123
6.2.1 Estimates for the velocity field 123
6.2.2 Estimates for pressure 127
6.2.3 Estimates for stresses 128
6.2.4 Estimates in combined norms 128
6.2.5 Lower bounds of errors 130
6.2.6 Mixed boundary conditions 131
6.2.7 Problems for almost incompressible fluids 137
6.2.8 Problems with the condition div u = p 139
6.3 Generalized Stokes problem 140
6.3.1 Estimates for solenoidal approximations 141
6.3.2 Estimates for nonsolenoidal fields 145
6.3.3 Estimates for the pressure field 146
6.3.4 Error minorant 148
6.3.5 Models with polymerization 148
6.3.6 Models with rotation 149
6.4 The Oseen problem 151
6.5 Stationary Navier-Stokes problem for d = 2 153
6.6 Notes for the chapter 156
7 Generalizations 158
7.1 Linear elliptic problem 158
7.1.1 The variational method 159
7.1.2 The method of integral identities 164
7.1.3 Error estimates for the dual variable 168
7.1.4 Two-sided estimates for combined norms 168
7.2 Elliptic problems with lower terms 171
7.3 Problems with solutions defined in subspaces 173
7.3.1 Abstract problem 173
7.3.2 Estimate for approximations lying in the subspace 173
7.3.3 Estimate for approximations lying in the energy space . 174
7.4 Derivation of a posteriori estimates from saddle point relations . 176
8 Nonlinear problems 178
8.1 Variational inequalities 178
8.1.1 Variational inequalities of the first kind 179
8.1.2 Variational inequalities of the second kind 185
8.2 General elliptic problem. Variational method 186
8.3 General elliptic problem. Nonvariational method 191
8.4 A posteriori estimates for special classes of nonlinear elliptic problems 196
8.4.1 a-Laplacian 196
8.4.2 Problems with nonlinear boundary conditions 201
8.4.3 Generalized Newtonian fluids 211
8.5 Notes for the chapter 214
9 Other problems 218
9.1 Differential equations of higher order 218
9.2 Equations with the operator curl 224
9.3 Evolutionary problems 229
9.3.1 The linear evolutionary problem 229
9.3.2 First form of the error majorant 231
9.3.3 Second form of the error majorant 235
9.3.4 Equivalence of the deviation and majorant 238
9.3.5 Comments 240
9.4 A posteriori estimates for optimal control problems 242
9.4.1 Two-sided bounds for cost functionals 243
9.4.2 Estimates for state and control functions 248
9.4.3 Estimate in a combined norm 251
9.4.4 Generalizations 252
9.4.5 Comments 254
9.5 Estimates for nonconforming approximations 254
9.5.1 Estimates based on projecting to the energy space 255
9.5.2 Estimates based on the Helmholtz decomposition 257
9.5.3 Accuracy of approximations obtained by the Trefftz method . 263
9.5.4 Comments 264
9.6 Uncertain data 265
9.6.1 Introduction 265
9.6.2 Errors caused by indeterminacy in coefficients 270
9.6.3 Errors owing to uncertain Q 276
9.6.4 Comments 278
9.7 Error estimates in terms of functionals and nonenergy norms 279
9.7.1 General framework 279
9.7.2 Estimates in local norms 280
9.7.3 Estimates in terms of linear functionals 281
9.7.4 Estimates based on the Poincare inequality 284
9.7.5 Estimates based on multiplicative inequalities 285
9.7.6 Estimates based on the maximum principle 285
9.7.7 Estimates in weighted norms 287
Bibliography 291
Index 314 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T19:39:01Z |
| indexdate | 2024-07-09T21:10:40Z |
| institution | BVB |
| isbn | 9783110191530 |
| language | English |
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| physical | XI, 316 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | de Gruyter |
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| series | Radon series on computational and applied mathematics |
| series2 | Radon series on computational and applied mathematics |
| spelling | Repin, Sergej I. Verfasser aut A posteriori estimates for partial differential equations Sergey Repin Berlin [u.a.] de Gruyter 2008 XI, 316 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Radon series on computational and applied mathematics 4 Differential equations, Partial Error analysis (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf A-posteriori-Abschätzung (DE-588)4346907-3 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s A-posteriori-Abschätzung (DE-588)4346907-3 s DE-604 Radon series on computational and applied mathematics 4 (DE-604)BV023335470 4 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016288519&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Repin, Sergej I. A posteriori estimates for partial differential equations Radon series on computational and applied mathematics Differential equations, Partial Error analysis (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd A-posteriori-Abschätzung (DE-588)4346907-3 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4346907-3 |
| title | A posteriori estimates for partial differential equations |
| title_auth | A posteriori estimates for partial differential equations |
| title_exact_search | A posteriori estimates for partial differential equations |
| title_exact_search_txtP | A posteriori estimates for partial differential equations |
| title_full | A posteriori estimates for partial differential equations Sergey Repin |
| title_fullStr | A posteriori estimates for partial differential equations Sergey Repin |
| title_full_unstemmed | A posteriori estimates for partial differential equations Sergey Repin |
| title_short | A posteriori estimates for partial differential equations |
| title_sort | a posteriori estimates for partial differential equations |
| topic | Differential equations, Partial Error analysis (Mathematics) Partielle Differentialgleichung (DE-588)4044779-0 gnd A-posteriori-Abschätzung (DE-588)4346907-3 gnd |
| topic_facet | Differential equations, Partial Error analysis (Mathematics) Partielle Differentialgleichung A-posteriori-Abschätzung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016288519&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023335470 |
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