Preface Chapter 1 Variational Formulation of Elliptic Problems 1.1 Basic concepts of Sobolev space 1.2 Variational formulation 1.3 Exercises Chapter 2 Finite Element Methods for Elliptic Equations 2.1 Galerkin method for variational problems 2.2 The construction of finite element spaces 2.2.1 The finite element 2.3 Computational consideration 2.4 Exercises Chapter 3 Convergence Theory of Finite Element Methods 3.1 Interpolation theory in Sobolev spaces 3.2 The energy error estimate 3.3 The L^2error estimate 3.4 Exercises Chapter 4 Adaptive Finite Element Methods 4.1 An example with singularity 4.2 A posteriori error analysis 4.2.1 The Clement interpolation operator 4.2.2 A posteriori error estimates 4.3 Adaptive algorithm 4.4 Convergence analysis 4.5 Exercises Chapter 5 Finite Element Multigrid Methods 5.1 The model problem 5.2 Iterative methods 5.3 The multigrid V-cycle algorithm 5.4 The finite element multigrid V-cycle algorithm 5.5 The full multigrid and work estimate 5.6 The adaptive multigrid method 5.7 Exercises Chapter 6 Mixed Finite Element Methods 6.1 Abstract framework 6.2 The Poisson equation as a mixed problem 6.3 The Stokes problem 6.4 Exercises Chapter 7 Finite Element Methods for Parabolic Problems 7.1 The weak solutions of parabolic equations 7.2 The semidiscrete approximation 7.3 The fully discrete approximation 7.4 A posteriori error analysis 7.5 The adaptive algorithm 7.6 Exercises Chapter 8 Finite Element Methods for Maxwell Equations 8.1 The function space H(curl;Ω) 8.2 The curl conforming finite element approximation 8.3 Finite element methods for time harmonic Maxwell equations 8.4 A posteriori error analysis 8.5 Exercises Chapter 9 Multiscale Finite Element Methods for Elliptic Equations 9.1 The homogenization result 9.2 The multiscale finite element method 9.2.1 Error estimate when h<ε 9.2.2 Error estimate when h>ε 9.3 The over-sampling multiscale finite element method 9.4 Exercises Chapter 10 Implementations 10.1 A brief introduction to the MATLAB PDE Toolbox 10.1.1 A first examplePoisson equation on the unit disk 10.1.2 The mesh data structure 10.1.3 A quick reference 10.2 Codes for Example 4.1L-shaped domain problem on uniform meshes 10.2.1 The main script 10.2.2 H1 error 10.2.3 Seven-point Gauss quadrature rule 10.3 Codes for Example 4.6|L-shaped domain problem on adaptive meshes 10.4 Implementation of the multigrid V-cycle algorithm 10.4.1 Matrix versions for the multigrid V-cycle algorithm and FMG 10.4.2 Code for FMG 10.4.3 Code for the multigrid V-cycle algorithm 10.4.4 The'newest vertex bisection'algorithm for mesh refinements 10.5 Exercises Bibliography
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