This definitive introduction to finite element methods has
been thoroughly updated for this third edition， which features
important new material for both research and application of the
finite element method.
The discussion of saddle point problems is a lughlight of the
book and has been elaborated to include many more nonstandard
applications. The chapter on applications in elasticity now
contains a complete discussion of locking phenomena.
The numerical solution ofelliptic partial differential equations
is an important application of finite elements and the author
discusses this subject comprehensively. These equations are treated
as variational problems for which the Sobolev spaces are the right
framework. Graduate students who do not necessarily have any
particular background in differential equations but require an
introduction to finite element methods will find this text
invaluable. Specifically， the chapter on finite elements in solid
mechanics provides a bridge between mathematics and
engineering.
目錄：
Preface to the Third English Edition
Preface to the First English Edition
Preface to the German Edition
Notation
Chapter Ⅰ Introduction
1. Examples and Classification of PDE''s
Examples
Classification of PDE''s
Well-posed problems
Problems
2. The Maximum Ptinciple
Examples
Corollaries
Problem
3. Finite Difference Methods
Discretization
Discrete maximum principle
Problem
4. A Convergence Theory for Difference Methods
Consistency
Local and global error
Limits of the con-vergence theory
Ptoblems
Chapter Ⅱ Conforming Finite Elements
1. Sobolev Spaces
Introduction to Sobolev spaces
Friedrichs'' inequality
Possible singularities of H1 functions
Compact imbeddings
Problems
2. Variational Formulation of Elliptic Boundary-Value Problems of
Second Order
Variational formulation
Reduction to homogeneous bound- ary conditions
Existence of solutions
Inhomogeneous boundary conditions
Problems
3. The Neumann Boundary-Value Problem. A Trace Theorem
Ellipticity in H
Boundary-value problems with natural bound-ary conditions
Neumann boundary conditions
Mixed boundary conditions
Proof of the trace theorem
Practi- cal consequences of the trace theorem
Problems
4. The Ritz-Galerkin Method and Some Finite Elements
Model problem
Problems
5. Some Standard Finite Elements
Requirements on the meshes
Significance of the differentia-bility properties
Triangular elements with complete polyno-mials
Remarks on Cl elements
Bilinear elements
Quadratic rectangular elements
Affine families
Choiceof an element
Problems
6. Approximation Properties
The Bramble-Hilbert lemma
Triangular elements with com-plete polynomials
Bilinear quadrilateral elements
In-verse estimates
Clement''s interpolation
Appendix: On the optimality of the estimates
Problems
7. Error Bounds for Elliptic Problems of Second Order
Remarks on regularity
Error bounds in the energy normL2 estimates
A simple Loo estimate
The L2-projector
Problems
8. Computational Considerations
Assembling the stiffness matrix
Static condensation
Complexity of setting up the matrix
Effect on the choice of a grid
Local mesh refinement
Implementation of the Neumann boundary-value problem
Problems
Chapter Ⅲ Nonconforming and Other Methods
1. Abstract Lemmas and a Simple Boundary Approximation
Generalizations of Cea''s lemma
Duality methods
The Crouzeix-Raviart element
A simple approximation to curved boundaries
Modifications of the duality argument
Problems
2. Isoparametric Elements
Isoparametric triangular elements
Isoparametric quadrilateral elements
Problems
3. Further Tools from Functional Analysis
Negative norms
Adjoint operators
An abstract exis- tence theorem
An abstract convergence theorem
Proof of Theorem 3.4
Problems
4. Saddle Point Problems
Saddle points and minima
The inf-sup condition
Mixed finite element methods
Fortin interpolation
……
Chapter Ⅳ The Conjugate Gradient Method
Chapter Ⅴ Multigrid Methods
Chapter Ⅵ Finite Elements in Solid Mechanics
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