有限元方法的數(shù)學理論

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.

內容概要

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.

書籍目錄

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

章節(jié)摘錄

　　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.）　　......

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