| 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 |
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