Language: English
ISBN/ISSN: 7030283382
Published on: 2010-01
Hardcover
This book grows out of the lectures the first author gave in the summer of 2002 in the Institute of Computational Mathematics of Chinese Academy of Sciences. The purpose of the lectures was to present a concise introduction to the basic ideas and mathematical tools in the construction and analysis of finite element methods for solving partial differential equations so that the students can start to do research on the theory and applications of the finite element method after the summer course. Some of the materials of the book have been taught several times by the authors in Nanjing University and Peking University. The current form of the book is based on the lecture notes which are constantly updated and expanded reflecting the newest development of the topics through the years.
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 l2 error 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 example--poisson equation on the unit disk
10.1.2 the mesh data structure
10.1.3 a quick reference
10.2 codes for example 4.1--l-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 inultigrid v-cycle algorithm
10.4.4 the "newest vertex bisection" algorithm for mesh refinements
10.5 exercises
bibliography
