Adaptive moving mesh methods:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2011
|
| Series: | Applied Mathematical Sciences
174 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 409-425 |
| Physical Description: | XVII, 432 S. Ill., graph. Darst. |
| ISBN: | 9781441979155 |
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Record in the Search Index
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| adam_text |
Titel: Adaptive moving mesh methods
Autor: Huang, Weizhang
Jahr: 2011
Contents
1 Introduction. 1
1.1 A model problem. 1
1.2 A moving finite difference method. 2
1.2.1 Finite difference method on a fixed mesh. 2
1.2.2 Finite difference method on an adaptive moving mesh. 3
1.3 A moving finite element method. 7
1.3.1 Finite element method on a fixed mesh. 7
1.3.2 Finite element method on an adaptive moving mesh. 11
1.4 Burgers' equation with an exact solution. 14
1.5 Basic components of a moving mesh method . 17
1.5.1 Mesh movement strategies . 18
1.5.2 Discretization of PDEs on a moving mesh. 18
1.5.3 Simultaneous or alternate solution. 20
1.6 Biographical notes. 21
1.7 Exercises. 23
2 Adaptive Mesh Movement in ID. 27
2.1 The equidistribution principle. 28
2.1.1 Equidistribution. 28
2.1.2 Optimality of equidistribution. 30
2.1.3 Equidistributing meshes as uniform meshes in a metric space 34
2.1.4 Another view of equidistribution_. 34
2.2 Computation of equidistributing meshes . 36
2.2.1 De Boor's algorithm. 36
2.2.2 BVP method. 40
2.3 Moving mesh PDEs . 43
2.3.1 MMPDEs in terms of coordinate transformation. 43
2.3.2 MMPDEs in terms of inverse coordinate transformation_ 50
Contents
2.4 Mesh density functions based on interpolation error. 53
2.4.1 Interpolation error estimates. 54
2.4.2 Optimal mesh density functions. 56
2.4.3 Error bounds for commonly used non-optimal mesh
density functions. 64
2.4.4 Summary of mesh density functions and error bounds. 66
2.4.5 Error bounds for a function with boundary layer. 69
2.5 Computation of mesh density functions and examples. 74
2.5.1 Recovery of solution derivatives. 74
2.5.2 Smoothing of mesh density functions and smoothed
MMPDEs . 76
2.5.3 Mesh density functions for solutions with multicomponents. 81
2.5.4 Examples with analytical functions. 81
2.6 Alternate solution procedures . 85
2.6.1 Alternate solution with quasi-Lagrange treatment of mesh
movement. 87
2.6.2 Rezoning treatment of mesh movement. 96
2.6.3 Interpolation on moving meshes. 97
2.7 Examples of applications. 99
2.8 Mesh density functions based on scaling invariance.Ill
2.8.1 Dimensional analysis, scaling invariance, and dominance
of equidistribution .114
2.8.2 MMPDE5 with constant ô.116
2.8.3 MMPDE5 with variable ô.119
2.8.4 Numerical results.119
2.9 Mesh density functions based on a posteriori error estimates.120
2.9.1 An a priori error estimate.123
2.9.2 An a posteriori error estimate.124
2.9.3 Optimal mesh density function and convergence results_125
2.9.4 Iterative algorithm for computing equidistributing meshes
and numerical examples .127
2.10 Biographical notes.130
2.11 Exercises .I33
Discretization of PDEs on Time-Varying Meshes.137
3.1 Coordinate transformations.138
3.1.1 Coordinate transformation as a mesh.138
3.1.2 Transformation relations.138
3.1.3 Transformed structure of PDEs.I44
3.1.4 Transformation relations in 2D.145
3.2 Finite difference methods.I47
Contents xv
3.2.1 The quasi-Lagrange approach.148
3.2.2 The rezoning approach.156
3.3 Finite element methods.157
3.3.1 Concepts of unstructured meshes and finite elements .157
3.3.2 Simplicial elements and cf-simplexes.165
3.3.3 The quasi-Lagrange approach.166
3.3.4 The rezoning approach.172
3.4 Two-mesh strategy for mesh movement.172
3.5 Interpolation on moving meshes.173
3.5.1 Linear interpolation.174
3.5.2 PDE-based interpolation.174
3.6 Biographical notes.175
3.7 Exercises .176
4 Basic Principles of Multidimensional Mesh Adaptation.177
4.1 Mesh adaptation from perspective of uniform meshes in a metric
space.178
4.1.1 Mathematical description of M-uniform meshes .179
4.1.2 Equidistribution and alignment conditions.180
4.2 Mesh control perspective.186
4.2.1 Jacobian matrix and size, shape, and orientation of mesh
elements.187
4.2.2 Mesh adaptation via metric specification.190
4.2.3 Geometric interpretations of mesh equidistribution and
alignment.193
4.2.4 Special case: scalar monitor functions.195
4.3 Continuous perspective.196
4.4 Function approximation perspective.200
4.5 Mesh quality measures.202
4.6 Analytical and numerical examples.208
4.7 Biographical notes.211
4.8 Exercises .213
5 Monitor Functions.215
5.1 Interpolation theory in Sobolev spaces.216
. 1 Error estimates for linear Lagrange interpolation at vertices .216
.2 A classical result.220
.3 Relations between norms on affine-equivalent elements_222
.4 Isotropie error bounds.228
.5 Anisotropie error bounds: Case / = 1.230
.6 Anisotropie error bounds: Case / 2.231
Contents
5.1.7 Interpolation error on element faces.231
5.2 Monitor functions based on interpolation error.234
5.2.1 Monitor function based on isotropic error estimates.234
5.2.2 Monitor function based on anisotropic error estimates: / = 1 246
5.2.3 Monitor function based on anisotropic error estimates: 1 — 2 253
5.2.4 The Hessian as the monitor function.262
5.2.5 Summary of formulas - continuous form.265
5.3 Computation of monitor functions .266
5.3.1 Recovery of solution derivatives.266
5.3.2 Computation of the absolute value of Hessian matrix.267
5.3.3 Smoothing.272
5.3.4 Monitor functions for multicomponent solutions.272
5.4 Monitor functions based on semi-a posteriori and a posteriori
error estimates.273
5.4.1 A semi-a posteriori method .274
5.4.2 A hierarchical basis method.276
5.5 Additional considerations for defining monitor functions.278
5.5.1 Monitor functions based on distance to interfaces.278
5.5.2 Monitor functions based on a reference mesh.278
5.6 Biographical notes.280
5.7 Exercises .280
Variational Mesh Adaptation Methods .281
6.1 General framework for variational methods and MMPDEs.282
6.1.1 General adaptation functional and mesh equations.283
6.1.2 Moving mesh PDEs.294
6.1.3 Boundary conditions for coordinate transformation.297
6.2 Existence of minimizer.298
6.2.1 Convex functional.299
6.2.2 Polyconvex functional.301
6.2.3 Examples of convex and polyconvex mesh adaptation
functionals.303
6.3 Discretization and solution procedures.306
6.3.1 Finite difference methods.307
6.3.2 Finite element methods.3Ð
6.4 Methods based on equidistribution and alignment conditions.312
6.4.1 Functional for mesh alignment.312
6.4.2 Functional for equidistribution.313
6.4.3 Mesh adaptation functional.314
6.4.4 Another mesh adaptation functional.317
6.4.5 Numerical examples.320
Contents xvii
6.5 Methods based on physical and geometric models.325
6.5.1 Variable diffusion methods.326
6.5.2 Harmonic mapping methods.337
6.5.3 Hybrid methods and directional control.345
6.5.4 Jacobian-weighted methods .349
6.5.5 Methods based on mechanical models.352
6.5.6 Methods based on Monge-Ampère equation /
Monge-Kantorovich optimal transport problem.356
6.5.7 Summary.362
6.6 Examples of applications.364
6.7 Biographical notes.371
6.8 Exercises .372
7 Velocity-Based Adaptive Methods.379
7.1 Methods based on geometric conservation law.379
7.1.1 GCL method.380
7.1.2 Deformation map method.387
7.1.3 Static version .387
7.1.4 A moving mesh finite element method based on GCL.388
7.2 MFE - moving finite element method.392
7.3 Other approaches.395
7.3.1 Method based on attraction-repulsion.395
7.3.2 Methods based on spring models.396
7.3.3 Methods based on minimizing convection terms .398
7.4 Exercises.399
A Sobolev spaces.401
? Arithmetic-mean geometric-mean inequality and Jensen's inequality 407
References.409
Nomenclature.427
Index.429 |
| any_adam_object | 1 |
| author | Huang, Weizhang Russell, Robert D. |
| author_GND | (DE-588)142830380 (DE-588)113926014 |
| author_facet | Huang, Weizhang Russell, Robert D. |
| author_role | aut aut |
| author_sort | Huang, Weizhang |
| author_variant | w h wh r d r rd rdr |
| building | Verbundindex |
| bvnumber | BV036779824 |
| classification_rvk | SK 540 SK 920 |
| ctrlnum | (OCoLC)700576290 (DE-599)DNB1006927425 |
| discipline | Mathematik |
| format | Book |
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| spelling | Huang, Weizhang Verfasser (DE-588)142830380 aut Adaptive moving mesh methods Weizhang Huang ; Robert D. Russell New York [u.a.] Springer 2011 XVII, 432 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied Mathematical Sciences 174 Literaturverz. S. 409-425 Gitterverfeinerung (DE-588)4482690-4 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Zeitabhängigkeit (DE-588)4320088-6 gnd rswk-swf Gittererzeugung (DE-588)4402542-7 gnd rswk-swf Adaptives Gitter (DE-588)4333769-7 gnd rswk-swf Adaptives Gitter (DE-588)4333769-7 s Partielle Differentialgleichung (DE-588)4044779-0 s Gitterverfeinerung (DE-588)4482690-4 s Gittererzeugung (DE-588)4402542-7 s Zeitabhängigkeit (DE-588)4320088-6 s DE-604 Russell, Robert D. Verfasser (DE-588)113926014 aut Erscheint auch als Online-Ausgabe 978-1-4419-7916-2 Applied Mathematical Sciences 174 (DE-604)BV000005274 174 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3536977&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020696508&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Huang, Weizhang Russell, Robert D. Adaptive moving mesh methods Applied Mathematical Sciences Gitterverfeinerung (DE-588)4482690-4 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Zeitabhängigkeit (DE-588)4320088-6 gnd Gittererzeugung (DE-588)4402542-7 gnd Adaptives Gitter (DE-588)4333769-7 gnd |
| subject_GND | (DE-588)4482690-4 (DE-588)4044779-0 (DE-588)4320088-6 (DE-588)4402542-7 (DE-588)4333769-7 |
| title | Adaptive moving mesh methods |
| title_auth | Adaptive moving mesh methods |
| title_exact_search | Adaptive moving mesh methods |
| title_full | Adaptive moving mesh methods Weizhang Huang ; Robert D. Russell |
| title_fullStr | Adaptive moving mesh methods Weizhang Huang ; Robert D. Russell |
| title_full_unstemmed | Adaptive moving mesh methods Weizhang Huang ; Robert D. Russell |
| title_short | Adaptive moving mesh methods |
| title_sort | adaptive moving mesh methods |
| topic | Gitterverfeinerung (DE-588)4482690-4 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Zeitabhängigkeit (DE-588)4320088-6 gnd Gittererzeugung (DE-588)4402542-7 gnd Adaptives Gitter (DE-588)4333769-7 gnd |
| topic_facet | Gitterverfeinerung Partielle Differentialgleichung Zeitabhängigkeit Gittererzeugung Adaptives Gitter |
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