Adaptive grid methods are among the most important classes of numerical methodsfor partial differential equations that arise from scientific and engineering computing.The study of this type of methods has been very active in recent years for algorithmdesign,theoretical analysis and applications to practical computations. This volumecontains a number of self-contained articles for adaptive finite element and finite dif-ference methods, which is aimed to provide some introduction materials for graduatestudents and junior researchers and a collection of references for researchers andpractitioners.

The articles in this volume,which are ordered alphabetically by authors,touchupon various aspects of adaptive methods for algorithmic design,theoretical analy-sis and practical applications.Chapter 1,by Long Chert and Jinchao Xu,summarizesthe basic techniques in local adaptive finite element methods and especially the recentadvancements on the convergence and complexity of this type of adaptive methods.Chapter 2,again by Long Chen and Jinchao Xu,describes various postprocessingtechniques for gradient recovery that include results on patch symmetric grids,mildlystructured grids and general unstructured grids.

Chapter 1 Convergence of Adaptive Finite Element Methods

 1.1 Introduction

 1.2 Preliminaries

 1.3 Residual type error estimator

 1.4 Convergence of an adaptive finite element method

 1.5 Optimality of the adaptive finite element method

Bibliography

Chapter 2 A Posteriori Error Estimator by Post-processing

 2.1 Introduction

 2.2 Linear finite element on patch symmetric grids

 2.3 Linear finite element on mildly structured grids

 2.4 Linear finite element on general unstructured grids

Bibliography

Chapter 3 Anisotropic Mesh Adaptation and Movement

 3.1 Introduction

3.1.1 Sobolev spaces

3.1.2 Mesh terminology

3.1.3 Two algebraic inequalities

 3.2 Basic principles in mesh adaptation

3.2.1 Geometric meaning of SVD decomposition

3.2.2 Alignment and equidistribution

3.2.3 Alignment and equidistribution for finite element meshes

 3.3 Interpolation theory in Sobolev spaces

3.3.1 Finite element terminology

3.3.2 Element-wise estimate on interpolation error

 3.4 Isotropic error estimates

3.4.1 Chain rule

3.4.2 Isotropic error estimation on a general mesh

3.4.3 Error bound on regular triangulations

 3.5 Anisotropic error estimates

3.5.1 An anisotropic error bound

3.5.2 Anisotropic error estimates independent of coordinate system

3.5.3 Bibliographic notes

 3.6 Mesh quality measures and monitor functions

3.6.1 Mesh quality measures

3.6.2 The isotropic case

3.6.3 The anisotropic case: ι=1

3.6.4 The anisotropic case: ι=2

 3.7 Anisotropic mesh adaptation: Refinement approach

3.7.1 Metric tensor

3.7.2 Numerical experiments

 3.8 Anisotropic mesh adaptation: Variational approach

3.8.1 Functional for mesh alignment

3.8.2 Functional for equidistribution

3.8.3 Mesh adaptation functional

3.8.4 Mesh equation

3.8.5 Numerical experiments

 3.9 Adaptive moving mesh methods: MMPDE approach

3.9.1 The MMPDE method

3.9.2 Numerical examples

 3.10 Adaptive moving mesh methods: GCL approach

3.10.1 GCL method

3.10.2 Relation to the Lagrange method and the deformation map method

3.10.3 Choice of ψ,νref,and ρ

3.10.4 Numerical examples

 3.11 Conclusions

  Bibliography

Chapter 4 Convergence Theory of Moving Grid Methods

 4.1 Introduction

 4.2 Maximum norm a posteriori error estimates for a ld singularly perturbed convection-diffusion equation

4.2.1 Model convection-diffusion problem

4.2.2 Numerical methods and notation

4.2.3 Stability properties of the differential operator

4.2.4 First-order error estimates

4.2.5 Second-order error estimates

 4.3 Full analysis of a robust adaptive method for a ld convection-diffu-sion problem

4.3.1 Upwind difference scheme

4.3.2 Adaptive mesh movement by equidistribution of the arc-length monitor function

4.3.3 The algorithm

4.3.4 The existence theorem

4.3.5 Accuracy of equidistributed and computed solutions

4.3.6 How many iterations for s-uniform accuracy?

4.3.7 Numerical results

4.3.8 Possible generalizations

Bibliography

Chapter 5 Computation of Crystalline Mierostructures with The Mesh Transformation Method

 5.1 Introduction

5.1.1 An illustrative example

5.1.2 The idea of the mesh transformation method (MTM)

 5.2 Mesh transformation and regular re-meshing

5.2.1 Some numerical examples

 5.3 Regularized mesh transformation methods

5.3.1 Existence and convergence theorems for RMT

5.3.2 Regularized periodic relaxation method (RPR)

 5.4 Application in computing non-smooth minimizers

5.4.1 Discrete relaxation problem

5.4.2 Numerical results

Bibliography

Chapter 6 On The Use of Moving Mesh Methods to Solve PDEs

 6.1 Introduction

 6.2 Moving mesh partial differential equations (MMPDEs)

6.2.1 Mesh equidistribution

6.2.2 Variational formulation

 6.3 Phase change problems

6.3.1 Convective heat transfer

6.3.2 Phase-field models

 6.4 A one-dimensional hr-adaptive method

6.4.1 The r-refinement strategy

6.4.2 h-refinement strategy

6.4.3 Monitor function

6.4.4 Refinement formula

6.4.5 Outline of hr-adaptive algorithm

6.4.6 Numerical experiments

 6.5 Conclusions

Bibliography

Chapter 7 Theory and Application of Adaptive Moving Grid Methods

 7.1 Introduction

 7.2 Adaptive moving grids in one dimension

7.2.1 A simple boundary-value problem

7.2.2 The equidistribution principle

7.2.3 Smoothing of the grid in space and time

7.2.4 Applications

 7.3 The higher-dimensional case

7.3.1 A tensor-grid approach in 2D

7.3.2 Smooth adaptive grids based on Winslow's approach

7.3.3 Three-dimensional adaptive moving grids

Bibliography

Chapter 8 Recovery Techniques in Finite Element Methods

 8.1 Introduction andprelirninary

 8.2 Local recovery for ID

8.2.1 Motivation-linear element

8.2.2 Higher order elements

8.2.3 One dimensional theoretical results

 8.3 Local recovery in higher dimensions

8.3.1 Methods and examples

8.3.2 Properties of the gradient recovery operator

8.3.3 Superconvergence analysis in 2D

 8.4 Quasi-local,semi-local,and global recoveries

Bibliography

Index