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 P-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 Wp1 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
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