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本书源于第一作者在2002年夏季在中国科学院计算数学研究所的演讲。这些演讲的目的是简要介绍构造和计算中的基本思想和数学工具。 求解偏微分方程的有限元方法分析，使学生可以在暑期课程之后开始进行有限元方法的理论和应用研究。本书的某些材料已被作者多次在南京大学和北京大学讲授。本书的当前形式基于讲义，这些讲义不断更新和扩展，以反映多年来主题的最新发展。

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• Contents
  Preface
  Chapter 1 Variational Formulation of Elliptic Problems 1
  1.1 Basic concepts of Sobolev space 1
  1.2 Variational formulation 8
  1.3 Exercises 11
  Chapter 2 Finite Element Methods for Elliptic Equations 13
  2.1 Galerkin method for variational problems 13
  2.2 The construction of finite element spaces 15
  2.2.1 The finite element 15
  2.3 Computational consideration 20
  2.4 Exercises 24
  Chapter 3 Convergence Theory of Finite Element Methods 25
  3.1 lnterpolation theory in Sobolev spaces 25
  3.2 The energy error estimate 31
  3.3 The L2 error estimate 32
  3.4 Exercises 33
  Chapter 4 Adaptive Finite Element Methods 34
  4.1 An example with singnlarity 34
  4.2 A posteriori error malysis 36
  4.2.1 The Clement interpolation operator 36
  4.2.2 A Posteriori error estimate 39
  4.3 Adaptive algorithm 41
  4.4 Convergence analysis 42
  4.5 Exercises 46
  Chapter 5 Finite Element Multigrid Methods 47
  5.1 The model problem 47
  5.2 Iterative methods 48
  5.3 The multigrid V-cycle algorithm.51
  5.4 The finite elemest multigrid V-cycle algorithm 57
  5.5 The full multigrid and work ostimate 58
  5.6 The adaptive multigrid method 59
  5.7 Exercises 60
  Chapter 6 Mixed Finite Element Methods 61
  6.1 Abstract framework 61
  6.2 The Poisson equation as a mixed problem 66
  6.3 The Stokes problem 70
  6.4 Exercises.73
  Chapter 7 Finite Element Methods for Parabolic Problems 74
  7.1 The wosk solutions of parabolic equations 74
  7.2 The semidiscrete approximation 78
  7.3 The fu11y discrete approximation 82
  7.4 A posteriori error oslalysis 86
  7.5 The adaptive algorithm 92
  7.6 Exercaces 97
  Chapter 8 Finite Element Methods for Maxwell Equations 98
  8.1 The function space H(curl；Ω) 99
  8.2 The curl conforming finite element approximation 106
  8.3 Finite element methods for time harmonic Maxwell equations 111
  8.4 A posteriori error oslalysis 114
  8.5 Exercises 120
  Chapter 9 Multisca1e Finite Element Methods for Elliptic Equations 122
  9.1 The homogenimtion resu1t 122
  9.2 The multiscale finite element method 126
  9.2.1 Error estimate when h＜ε 126
  9.2.2 Error estimate when h＞ε 128
  9.3 The over-sampling multiscale finite element method 131
  9.4 Exercises 137
  Chapter 10 Implementations 138
  10.1 A brief introduction to the MATLAB PDE Toolbox 138
  10.1.1 A first exam.ple-Poisson equation on the unit disk 139
  10.1.2 The mosh data structure 140
  10.1.3 A quick reference143
  10.2 Codes for Example 4.1-L-shaped domaln problem on uniform moshes 144
  10.2.1 The main script 144
  10.2.2 H error 145
  10.2.3 Seven-point Gauss quadrature rule 145
  10.3 Codes for Example 4.6-L-shaped domaln problemon adaptive moshes 146
  10.4 Implementation of the multigrid V-cycle algorithm 148
  10.4.1 Matrix versions for the multigrid V-cycle algorithm and FMG 148
  10.4.2 Code for FMG 149
  10.4.3 Code for the multigrid V-cycle algorithm 150
  10.4.4 The“newest vertex bisection”algorithm for mesh refinlosnents 152
  10.5 Exercises 158
  Bibliography 160