有限元方法的數(shù)學(xué)理論

內(nèi)容概要

有限元法被廣泛用于工程設(shè)計和工程分析。本書是Springer出版的《應(yīng)用數(shù)學(xué)教材》叢書之15。全書分成15章，在第1版的基礎(chǔ)上增加了加性Schwarz預(yù)條件和自適應(yīng)格；書中不但提供有限元法系統(tǒng)的數(shù)學(xué)理論。還兼重在工程設(shè)計和分析中的應(yīng)用算法效率、程序開發(fā)和較難的收斂問題。

書籍目錄

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

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