有限元方法的数学理论

章节摘录

　　We will take this opportunity to philosophize about some power-ful characteristics of the finite element formalism for generating discreteschemes for approximating the solutions to differential equations. Being based on the variational formulation of boundary value problems， it is quite systematic， handling different boundary conditions with ease； one simply re-places infinite dimensional spaces with finite dimensional subspaces. What results， as in （0.5.3）， is the same as a finite difference equation， in keeping with the dictum that different numerical methods are usually more similarthan they are distinct. However， we were able to derive very quickly the convergence properties of the finite element method. Finally， the notation for the discrete scheme is quite compact in the finite element for mulation.This could be utilized to make coding the algorithm much more efficient if only the appropriate computer language and compiler were available. Thislatter characteristic of the finite element method is one that has not yet been exploited extensively， but an initial attempt has been made in the sys-tem fec （Bagheri， Scott & Zhang 1992）. （One could also argue that finiteele ment practitioners have already taken advantage of this by developingtheir own "languages" through extensive software libraries of their own， but this applies equally well to the finite-difference practitioners.）　　......

前言

　　Mathematics is playing an ever more important role in the physical and biological sciences， provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas-sical techniques of applied mathematics. This renewal of interest， both inresearch and teaching， has led to the establishment of the series Texts in Applied Mathematics （TAM）.　　The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques， such asnumerical and symbolic computer systems， dynamical systems， and chaos，mix with and reinforce the traditional methods of applied mathematics.Thus， the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses.　　TAM will publish textbooks suitable for use in advanced undergraduateand beginning graduate courses， and will complement the Applied Mathe-matical Sciences （AMS） series， which will focus on advanced textbooks andresearch-level monographs.

书籍目录

series preface

preface to the third edition

preface to the second edition

preface to the first edition

0 basic concepts

0.1 weak formulation of boundary value problems

0.2 ritz-galerkin approximation

0.3 error estimates

0.4 piecewise polynomial spaces - the finite element method

0.5 relationship to difference methods

0.6 computer implementation of finite element methods

0.7 local estimates

0.8 adaptive approximation

0.9 weighted norm estimates

0.x exercises

1 sobolev spaces

1.1 review of lebesgue integration theory

1.2 generalized (weak) derivatives

1.3 sobolev norms and associated spaces

1.4 inclusion relations and sobolev's inequality

1.5 review of chapter 0

1.6 trace theorems

1.7 negative norms and duality

1.x exercises

2 variational formulation of elliptic boundary value problems

2.1 inner-product spaces

2.2 hilbert spaces

2.3 projections onto subspaces

2.4 riesz representation theorem

2.5 formulation of symmetric variational problems

2.6 formulation of nonsymmetric variational problems

2.7 the lax-milgram theorem

2.8 estimates for general finite element approximation

2.9 higher-dimensional examples

2.x exercises

3 the construction of a finite element space

3.1 the finite element

3.2 triangular finite elements

the lagrange element

the hermite element

the argyris element

3.3 the interpolant

3.4 equivalence of elements

3.5 rectangular elements

tensor product elements

the serendipity element

3.6 higher-dimensional elements

3.7 exotic elements

3.x exercises

4 polynomial approximation theory in sobolev spaces

4.1 averaged taylor polynomials

4.2 error representation

4.3 bounds for riesz potentials

4.4 bounds for the interpolation error

4.5 inverse estimates

4.6 tensor. product polynomial approximation

4.7 isoparametric polynomial approximation

4.8 interpolation of non-smooth functions

4.9 a discrete sobolev inequality

4.x exercises

5 n-dimensional variational problems

5.1 variational formulation of poisson's equation

5.2 variational formulation of the pure neumann problem

5.3 coercivity of the variational problem

5.4 variational approximation of poisson's equation

5.5 elliptic regularity estimates

5.6 general second-order elliptic operators

5.7 variational approximation of general elliptic problems

5.8 negative-norm estimates

5.9 the plate-bending biharmonic problem

5.x exercises

6 finite element multigrid methods

6.1 a model problem

6.2 mesh-dependent norms

6.3 the multigrid algorithm

6.4 approximation property

6.5 w-cycle convergence for the kth level iteration

6.6 ]/-cycle convergence for the kth level iteration

6.7 full multigrid convergence analysis and work estimates

6.x exercises

7 additive schwarz preconditioners

7.1 abstract additive schwarz framework

7.2 the hierarchical basis preconditioner

7.3 the bpx preconditioner

7.4 the two-level additive schwarz preconditioner

7.5 nonoverlapping domain decomposition methods

7.6 the bps preconditioner

7.7 the neumann-neumann preconditioner

7.8 the bddc preconditioner

7.x exercises

8 max-norm estimates

8.1 main theorem

8.2 reduction to weighted estimates

8.3 proof of lemma 8.2.6

8.4 proofs of lemmas 8.3.7 and 8.3.11

8.5 lp estimates (regular coefficients)

8.6 lp estimates (irregular coefficients)

8.7 a nonlinear example

8.x exercises

9 adaptive meshes

9.1 a priori estimates

9.2 error estimators

9.3 local error estimates

9.4 estimators for linear forms and other norms

9.5 a convergent adaptive algorithm

9.6 conditioning of finite element equations

9.7 bounds on the condition number

9.8 applications to the conjugate-gradient method

9.x exercises

10 variational crimes

10.1 departure from the framework

10.2 finite elements with interpolated boundary conditions

10.3 nonconforming finite elements

10.4 isoparametric finite elements

10.5 discontinuous finite elements

10.6 poincare-friedrichs inequalitites for piecewise w1p functions

10.x exercises

11 applications to planar elasticity

11.1 the boundary value problems

11.2 weak formulation and korn's inequality

11.3 finite element approximation and locking

11.4 a robust method for the pure displacement problem

11.x exercises

12 mixed methods

12.1 examples of mixed variational formulations

12.2 abstract mixed formulation

12.3 discrete mixed formulation

12.4 convergence results for velocity approximation

12.5 the discrete inf-sup condition

12.6 verification of the inf-sup condition

12.x exercises

13 iterative techniques for mixed methods

13.1 iterated penalty method

13.2 stopping criteria

13.3 augmented lagrangian method

13.4 application to the navier-stokes equations

13.5 computational examples

13.x exercises

14 applications of operator-interpolation theory

14.1 the real method of interpolation

14.2 real interpolation of sobolev spaces

14.3 finite element convergence estimates

14.4 the simultaneous approximation theorem

14.5 precise characterizations of regularity

14.x exercises

references

index

作者简介

《有限元方法的数学理论(第3版)》内容简介：This edition contains four new sections on the following topics: the BDDC domain decomposition preconditioner (Section 7.8), a convergent adaptive algorithm (Section 9.5), interior penalty methods (Section 10.5) and Poincare-Friedrichs inequalities for piecewise Wp1 functions (Section 10.6).We have made improvements throughout the text, many of which were suggested by colleagues, to whom we are grateful. New exercises have been added and the list of references has also been expanded and updated.

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