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Adaptive grid methods are among the most important classes of numerical methods for 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 algorithm design, theoretical analysis and applications to practical computations. This volume contains a number of self-contained articles for adaptive finite element and finite difference methods, which is aimed to provide some introduction materials for graduate students and junior researchers and a collection of references for researchers and practitioners.
These articles mostly grew out of the lectures notes that were given in Summer Workshops on Adaptive Method, Theory and Application organized by Tao Tang, Jinchao Xu and Pingwen Zhang in Peking University, China, during June 20--August 20, 2005. This summer school was aimed to provide a comprehensive and up-to-date presentation of modern theories and practical applications for adaptive computations. The main lecturers of the Summer School include Weizhang Huang of University of Kansas, Natalia Kopteva of University of Limerick, Zhiping Li of Peking University, John Mackenzie of Strathclyde University, Jinchao Xu of Penn State University, Paul Zegeling of Utrecht University, and Zhimin Zhang of Wayne State University. Other lecturers include Tao Tang, Xiaoping Wang of HKUST, Huazhong Tang and Pingwen Zhang (both from Peking University). More detailed information of this summer school can be found in http://ccse.pku.edu.cn/activities/2005/adaptiveseminar.htm(which is most in Chinese).

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• Chapter1 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 elementmethod
  Bibliography
  Chapter2 A Posteriori ErrorEstimator 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
  Chapter3 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 I nterpolation 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 90
  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:l=1
  3.6.4 The anisotropic case:l=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 w,vref,and ρ
  3.10.4 Numerical examples
  3.11 Conclusions
  Bibliography
  Chapter4 Convergence Theory of Moving Grid Methods
  4.1 Introduction
  4.2 Maximum norm a posteriori error estimates for a 1d 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 1d 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 forε-uniform accuracy?
  4.3.7 Numerical results
  4.3.8 Possible generalizations
  Bibliography
  Chapter5 Computation of Crystalline Microstructures with The Mesh Transformation Method
  5.1 Introduction
  5.1.1An 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
  Chapter6 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 Re.nement formula
  6.4.5 Outline of hr-adaptive algorithm
  6.4.6 Numerical experiments
  6.5 Conclusions
  Bibliography
  Chapter7 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 Thehigher-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
  Chapter8 Recovery Techniques in Finite Element Methods
  8.1 Introduction and preliminary
  8.2 Local recovery for 1D
  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