| Series Preface 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 o 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 Tile Lax-Milgram Theorem 2.8 Estimates for General Finite Element Approximation 2.9 Higher-dimensional Examples 2.10 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 3.8 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 Isoparanmtric Polynomial Approximation 4.8 Interpolation of Non-smooth Functions 4.9 A Discrete Sobolev Inequality 4.x Exercises 5 n-Dimensional Variational Problems 6 Finite Element Multigrid Methods 7 Additive Schwarz Preconditioners 8 Max-norm Estimates 9 Adaptive Meshes 10 Variational Crimes 11 Applications to Planar Elasticity 12 Mixed Methods 13 Iterative Techniques for Mixed Methods 14 Applications of Operator-Interpolation Theory References Index |
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