简介
Moving mesh methods are an effective, mesh-adaptation-based approach for the numerical solution of mathematical models of physical phenomena. Currently there exist three main strategies for mesh adaptation, namely, to use mesh subdivision,聽 local high order approximation (sometimes combined with mesh subdivision), and mesh movement. The latter type of adaptive mesh method has been less well studied, both computationally and theoretically. This book is about adaptive mesh generation and moving mesh methods for the numerical solution of time-dependent partial differential equations. It presents a general framework and theory for adaptive mesh generation and gives a comprehensive treatment of moving mesh methods and their basic components, along with their application for a number of nontrivial physical problems.聽 Many explicit examples with computed figures illustrate the various methods and the effects of parameter choices for those methods. The partial differential equations considered are mainly parabolic (diffusion-dominated, rather聽 than convection-dominated). The extensive bibliography provides an invaluable guide to the literature in this field. Each chapter contains useful exercises. Graduate students, researchers and practitioners working in this area will benefit from this book.聽Weizhang Huang is a Professor in the Department of Mathematics at the University of Kansas. Robert D. Russell is a Professor in the Department of Mathematics at Simon Fraser University.
目录
Introduction p. 1
A model problem p. 1
A moving finite difference method p. 2
Finite difference method on a fixed mesh p. 2
Finite difference method on an adaptive moving mesh p. 3
A moving finite element method p. 7
Finite element method on a fixed mesh p. 7
Finite element method on an adaptive moving mesh p. 11
Burgers'equation with an exact solution p. 14
Basic components of a moving mesh method p. 17
Mesh movement strategies p. 18
Discretization of PDEs on a moving mesh p. 18
Simultaneous or alternate solution p. 20
Biographical notes p. 21
Exercises p. 23
Adaptive Mesh Movement in ID p. 27
The equidistribution principle p. 28
Equidistribution p. 28
Optimality of equidistribution p. 30
Equidistributing meshes as uniform meshes in a metric space p. 34
Another view of equidistribution p. 34
Computation of equidistributing meshes p. 36
De Boor's algorithm p. 36
Bvp method p. 40
Moving mesh PDEs p. 43
MMPDEs in terms of coordinate transformation p. 43
MMPDEs in terms of inverse coordinate transformation p. 50
Mesh density functions based on interpolation enor p. 53
Interpolation error estimates p. 54
Optimal mesh density functions p. 56
Enor bounds for commonly used non-optimal mesh density functions p. 64
Summary of mesh density functions and error bounds p. 66
Error bounds for a function with boundary layer p. 69
Computation of mesh density functions and examples p. 74
Recovery of solution derivatives p. 74
Smoothing of mesh density functions and smoothed MMPDEs p. 76
Mesh density functions for solutions with multicomponents p. 81
Examples with analytical functions p. 81
Alternate solution procedures p. 85
Alternate solution with quasi-Lagrange treatment of mesh movement p. 87
Rezoning treatment of mesh movement p. 96
Interpolation on moving meshes p. 97
Examples of applications p. 99
Mesh density functions based on scaling invariance p. 111
Dimensional analysis, scaling invariance, and dominance of equidistribution p. 114
MMPDEs with constant p. 116
MMPDE5 with variable p. 119
Numerical results p. 119
Mesh density functions based on a posteriori error estimates p. 120
An a priori error estimate p. 123
An a posteriori error estimate p. 124
Optimal mesh density function and convergence results p. 125
Iterative algorithm for computing equidistributing meshes and numerical examples p. 127
Biographical noteS p. 130
Exercises p. 133
Discretization of PDEs on Time-Varying Meshes p. 137
Coordinate transformations p. 138
Coordinate transformation as a mesh p. 138
Transformation relations p. 138
Transformed structure of PDEs p. 144
Transformation relations in 2D p. 145
Finite difference methods p. 147
The quasi-Lagrange approach p. 148
The rezoning approach p. 156
Finite element methods p. 157
Concepts of unstructured meshes and finite elements p. 157
Simplicial elements and d-simplexes p. 165
The quasi-Lagrange approach p. 166
The rezoning approach p. 172
Two-mesh strategy for mesh movement p. 172
Interpolation on moving meshes p. 173
Linear interpolation p. 174
PDE-based interpolation p. 174
Biographical notes p. 175
Exercises p. 176
Basic Principles of Multidimensional Mesh Adaptation p. 177
Mesh adaptation from perspective of uniform meshes in a metric space p. 178
Mathematical description of M-uniform meshes p. 179
Equidistribution and alignment conditions p. 180
Mesh control perspective p. 186
Jacobian matrix and size, shape, and orientation of mesh elements p. 187
Mesh adaptation via metric specification p. 190
Geometric interpretations of mesh equidistribution and alignment p. 193
Special case: scalar monitor functions p. 195
Continuous perspective p. 196
Function approximation perspective p. 200
Mesh quality measures p. 202
Analytical and numerical examples p. 208
Biographical notes p. 211
Exercises p. 213
Monitor Functions p. 215
Interpolation theory in Sobolev spaces p. 216
Error estimates for linear Lagrange interpolation at vertices p. 216
A classical result p. 220
Relations between norms on affine-equivalent elements p. 222
Isotropic error bounds p. 228
Anisotropic error bounds: Case I=1 p. 230
Anisotropic error bounds: Case I>2 p. 231
Interpolation error on element faces p. 231
Monitor functions based on interpolation error p. 234
Monitor function based on isotropic error estimates p. 234
Monitor function based on anisotropic error estimates: I = 1 p. 246
Monitor function based on anisotropic error estimates: I = 2 p. 253
The Hessian as the monitor function p. 262
Summary of formulas-continuous form p. 265
Computation of monitor functions p. 266
Recovery of solution derivatives p. 266
Computation of the absolute value of Hessian matrix p. 267
Smoothing p. 272
Monitor functions for multicomponent solutions p. 272
Monitor functions based on semi-a posteriori and a posteriori error estimates p. 273
A semi-a posteriori method p. 274
A hierarchical basis method p. 276
Additional considerations for defining monitor functions p. 278
Monitor functions based on distance to interfaces p. 278
Monitor functions based on a reference mesh p. 278
Biographical notes p. 280
Exercises p. 280
Variational Mesh Adaptation Methods p. 281
General framework for variational methods and MMPDEs p. 282
General adaptation functional and mesh equations p. 283
Moving mesh PDEs p. 294
Boundary conditions for coordinate transformation p. 297
Existence of minimizer p. 298
Convex functionals p. 299
Polyconvex functionals p. 301
Examples of convex and polyconvex mesh adaptation functionals p. 303
Discretization and solution procedures p. 306
Finite difference methods p. 307
Finite element methods p. 311
Methods based on equidistribution and alignment conditions p. 312
Functional tor mesh alignment p. 312
Functional for equidistribution p. 313
Mesh adaptation functional p. 314
Another mesh adaptation functional p. 317
Numerical examples p. 320
Methods based on physical and geometric models p. 325
Variable diffusion methods p. 326
Harmonic mapping methods p. 337
Hybrid methods and directional control p. 345
Jacobian-weighted methods p. 349
Methods based on mechanical models p. 352
Methods based on Monge-Ampere equation Monge-Kantorovich optimal transport problem p. 356
Summary p. 362
Exaruples of applications p. 364
Biographical notes p. 371
Exercises p. 372
Velocity-Based Adaptive Methods p. 379
Methods based on geometric conservation law p. 379
GCLmethod p. 380
Deformation map method p. 387
Static version p. 387
A moving mesh finite element method based on GCL p. 388
MPE-moving finite element method p. 392
Other approaches p. 395
Method based on attraction-repulsion p. 395
Methods based on spring models p. 396
Methods based on minimizing convection tenus p. 398
Exercises p. 399
Soholev spaces p. 401
Arithmetic-mean geometric-mean inequality and Jensen's hiequaIity p. 407
References p. 409
Nomenelatnre p. 427
Index p. 429
A model problem p. 1
A moving finite difference method p. 2
Finite difference method on a fixed mesh p. 2
Finite difference method on an adaptive moving mesh p. 3
A moving finite element method p. 7
Finite element method on a fixed mesh p. 7
Finite element method on an adaptive moving mesh p. 11
Burgers'equation with an exact solution p. 14
Basic components of a moving mesh method p. 17
Mesh movement strategies p. 18
Discretization of PDEs on a moving mesh p. 18
Simultaneous or alternate solution p. 20
Biographical notes p. 21
Exercises p. 23
Adaptive Mesh Movement in ID p. 27
The equidistribution principle p. 28
Equidistribution p. 28
Optimality of equidistribution p. 30
Equidistributing meshes as uniform meshes in a metric space p. 34
Another view of equidistribution p. 34
Computation of equidistributing meshes p. 36
De Boor's algorithm p. 36
Bvp method p. 40
Moving mesh PDEs p. 43
MMPDEs in terms of coordinate transformation p. 43
MMPDEs in terms of inverse coordinate transformation p. 50
Mesh density functions based on interpolation enor p. 53
Interpolation error estimates p. 54
Optimal mesh density functions p. 56
Enor bounds for commonly used non-optimal mesh density functions p. 64
Summary of mesh density functions and error bounds p. 66
Error bounds for a function with boundary layer p. 69
Computation of mesh density functions and examples p. 74
Recovery of solution derivatives p. 74
Smoothing of mesh density functions and smoothed MMPDEs p. 76
Mesh density functions for solutions with multicomponents p. 81
Examples with analytical functions p. 81
Alternate solution procedures p. 85
Alternate solution with quasi-Lagrange treatment of mesh movement p. 87
Rezoning treatment of mesh movement p. 96
Interpolation on moving meshes p. 97
Examples of applications p. 99
Mesh density functions based on scaling invariance p. 111
Dimensional analysis, scaling invariance, and dominance of equidistribution p. 114
MMPDEs with constant p. 116
MMPDE5 with variable p. 119
Numerical results p. 119
Mesh density functions based on a posteriori error estimates p. 120
An a priori error estimate p. 123
An a posteriori error estimate p. 124
Optimal mesh density function and convergence results p. 125
Iterative algorithm for computing equidistributing meshes and numerical examples p. 127
Biographical noteS p. 130
Exercises p. 133
Discretization of PDEs on Time-Varying Meshes p. 137
Coordinate transformations p. 138
Coordinate transformation as a mesh p. 138
Transformation relations p. 138
Transformed structure of PDEs p. 144
Transformation relations in 2D p. 145
Finite difference methods p. 147
The quasi-Lagrange approach p. 148
The rezoning approach p. 156
Finite element methods p. 157
Concepts of unstructured meshes and finite elements p. 157
Simplicial elements and d-simplexes p. 165
The quasi-Lagrange approach p. 166
The rezoning approach p. 172
Two-mesh strategy for mesh movement p. 172
Interpolation on moving meshes p. 173
Linear interpolation p. 174
PDE-based interpolation p. 174
Biographical notes p. 175
Exercises p. 176
Basic Principles of Multidimensional Mesh Adaptation p. 177
Mesh adaptation from perspective of uniform meshes in a metric space p. 178
Mathematical description of M-uniform meshes p. 179
Equidistribution and alignment conditions p. 180
Mesh control perspective p. 186
Jacobian matrix and size, shape, and orientation of mesh elements p. 187
Mesh adaptation via metric specification p. 190
Geometric interpretations of mesh equidistribution and alignment p. 193
Special case: scalar monitor functions p. 195
Continuous perspective p. 196
Function approximation perspective p. 200
Mesh quality measures p. 202
Analytical and numerical examples p. 208
Biographical notes p. 211
Exercises p. 213
Monitor Functions p. 215
Interpolation theory in Sobolev spaces p. 216
Error estimates for linear Lagrange interpolation at vertices p. 216
A classical result p. 220
Relations between norms on affine-equivalent elements p. 222
Isotropic error bounds p. 228
Anisotropic error bounds: Case I=1 p. 230
Anisotropic error bounds: Case I>2 p. 231
Interpolation error on element faces p. 231
Monitor functions based on interpolation error p. 234
Monitor function based on isotropic error estimates p. 234
Monitor function based on anisotropic error estimates: I = 1 p. 246
Monitor function based on anisotropic error estimates: I = 2 p. 253
The Hessian as the monitor function p. 262
Summary of formulas-continuous form p. 265
Computation of monitor functions p. 266
Recovery of solution derivatives p. 266
Computation of the absolute value of Hessian matrix p. 267
Smoothing p. 272
Monitor functions for multicomponent solutions p. 272
Monitor functions based on semi-a posteriori and a posteriori error estimates p. 273
A semi-a posteriori method p. 274
A hierarchical basis method p. 276
Additional considerations for defining monitor functions p. 278
Monitor functions based on distance to interfaces p. 278
Monitor functions based on a reference mesh p. 278
Biographical notes p. 280
Exercises p. 280
Variational Mesh Adaptation Methods p. 281
General framework for variational methods and MMPDEs p. 282
General adaptation functional and mesh equations p. 283
Moving mesh PDEs p. 294
Boundary conditions for coordinate transformation p. 297
Existence of minimizer p. 298
Convex functionals p. 299
Polyconvex functionals p. 301
Examples of convex and polyconvex mesh adaptation functionals p. 303
Discretization and solution procedures p. 306
Finite difference methods p. 307
Finite element methods p. 311
Methods based on equidistribution and alignment conditions p. 312
Functional tor mesh alignment p. 312
Functional for equidistribution p. 313
Mesh adaptation functional p. 314
Another mesh adaptation functional p. 317
Numerical examples p. 320
Methods based on physical and geometric models p. 325
Variable diffusion methods p. 326
Harmonic mapping methods p. 337
Hybrid methods and directional control p. 345
Jacobian-weighted methods p. 349
Methods based on mechanical models p. 352
Methods based on Monge-Ampere equation Monge-Kantorovich optimal transport problem p. 356
Summary p. 362
Exaruples of applications p. 364
Biographical notes p. 371
Exercises p. 372
Velocity-Based Adaptive Methods p. 379
Methods based on geometric conservation law p. 379
GCLmethod p. 380
Deformation map method p. 387
Static version p. 387
A moving mesh finite element method based on GCL p. 388
MPE-moving finite element method p. 392
Other approaches p. 395
Method based on attraction-repulsion p. 395
Methods based on spring models p. 396
Methods based on minimizing convection tenus p. 398
Exercises p. 399
Soholev spaces p. 401
Arithmetic-mean geometric-mean inequality and Jensen's hiequaIity p. 407
References p. 409
Nomenelatnre p. 427
Index p. 429
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