有限元方法的数学理论

出版时间：2008-9  出版社：世界图书出版公司  作者：布雷  页数：361  
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内容概要

有限元法被广泛用于工程设计和工程分析。本书是Springer出版的《应用数学教材》丛书之15。全书分成15章，在第1版的基础上增加了加性Schwarz预条件和自适应格；书中不但提供有限元法系统的数学理论。还兼重在工程设计和分析中的应用算法效率、程序开发和较难的收敛问题。

书籍目录

Series PrefacePreface to the Second EditionPreface to the First Edition0 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 Exercises1 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   1.6 Trace Theorems  1.7 Negative Norms and Duality  1.x Exercises2 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 Exercises3 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 Exercises4 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 Exercises5 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 Exercises6 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 V-cycle Convergence for the kth Level Iteration  6.7 Full Multigrid Convergence Analysis and Work Estimates  6.x Exercises7 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.x Exercises8 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 Exercises9 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 Conditioning of Finite Element Equations  9.6 Bounds on the Condition Number  9.7 Applications to the Conjugate-Gradient Method  9.x Exercises10 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.x Exercises11 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 Exercises12 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 Exercises13 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 Exercises14 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 ExercisesReferencesIndex

用户评论 (总计2条)

•   书内容没的说，很不错。但是送到的书，外观上有损坏
•   别人推荐的，还没看，据说是有限元方面最经典的