Cover -- Title Page -- Copyright Page -- Preface -- Contents -- 1 Historical introduction -- 2 Weighted residual and variational methods -- 2.1 Classification of differential operators -- 2.2 Self-adjoint positive definite operators -- 2.3 Weighted residual methods -- 2.4 Extremum formulation: homogeneous boundary conditions -- 2.5 Non-homogeneous boundary conditions -- 2.6 Partial differential equations: natural boundary conditions -- 2.7 The Rayleigh-Ritz method -- 2.8 The 'elastic analogy' for Poisson's equation -- 2.9 Variational methods for time-dependent problems -- 2.10 Exercises and solutions -- 3 The finite element method for elliptic problems -- 3.1 Difficulties associated with the application of weighted residual methods -- 3.2 Piecewise application of the Galerkin method -- 3.3 Terminology -- 3.4 Finite element idealization -- 3.5 Illustrative problem involving one independent variable -- 3.6 Finite element equations for Poisson's equation -- 3.7 A rectangular element for Poisson's equation -- 3.8 A triangular element for Poisson's equation -- 3.9 Exercises and solutions -- 4 Higher-order elements: the isoparametric concept -- 4.1 A two-point boundary-value problem -- 4.2 Higher-order rectangular elements -- 4.3 Higher-order triangular elements -- 4.4 Two degrees of freedom at each node -- 4.5 Condensation of internal nodal freedoms -- 4.6 Curved boundaries and higher-order elements: isoparametric elements -- 4.7 Exercises and solutions -- 5 Further topics in the finite element method -- 5.1 The variational approach -- 5.2 Collocation and least squares methods -- 5.3 Use of Galerkin's method for time-dependent and non-linear problems -- 5.4 Time-dependent problems using variational principles which are not extremal -- 5.5 The Laplace transform -- 5.6 Exercises and solutions -- 6 Convergence of the finite element method.

6.1 A one-dimensional example -- 6.2 Two-dimensional problems involving Poisson's equation -- 6.3 Isoparametric elements: numerical integration -- 6.4 Non-conforming elements: the patch test -- 6.5 Comparison with the finite difference method: stability -- 6.6 Exercises and solutions -- 7 The boundary element method -- 7.1 Integral formulation of boundary-value problems -- 7.2 Boundary element idealization for Laplace's equation -- 7.3 A constant boundary element for Laplace's equation -- 7.4 A linear element for Laplace's equation -- 7.5 Time-dependent problems -- 7.6 Exercises and solutions -- 8 Computational aspects -- 8.1 Pre-processor -- 8.2 Solution phase -- 8.3 Post-processor -- 8.4 Finite element method (FEM) or boundary element method (BEM)? -- Appendix A Partial differential equation models in the physical sciences -- A.1 Parabolic problems -- A.2 Elliptic problems -- A.3 Hyperbolic problems -- A.4 Initial and boundary conditions -- Appendix B Some integral theorems of the vector calculus -- Appendix C A formula for integrating products of area coordinates over a triangle -- Appendix D Numerical integration formulae -- D.1 One-dimensional Gauss quadrature -- D.2 Two-dimensional Gauss quadrature -- D.3 Logarithmic Gauss quadrature -- Appendix E Stehfest's formula and weights for numerical Laplace transform inversion -- References -- Index.