In the wake of the computer revolution, a large number of apparently uncon­ nected computational techniques have emerged. Also, particular methods have assumed prominent positions in certain areas of application. Finite element methods, for example, are used almost exclusively for solving structural problems; spectral methods are becoming the preferred approach to global atmospheric modelling and weather prediction; and the use of finite difference methods is nearly universal in predicting the flow around aircraft wings and fuselages. These apparently unrelated techniques are firmly entrenched in computer codes used every day by practicing scientists and engineers. Many of these scientists and engineers have been drawn into the computational area without the benefit offormal computational training. Often the formal computational training we do provide reinforces the arbitrary divisions between the various computational methods available. One of the purposes of this monograph is to show that many computational techniques are, indeed, closely related. The Galerkin formulation, which is being used in many subject areas, provides the connection. Within the Galerkin frame-work we can generate finite element, finite difference, and spectral methods.

1. Traditional Galerkin Methods.- 1.1 Introduction.- 1.2 Simple Examples.- 1.3 Method of Weighted Residuals.- 1.4 Connection with Other Methods.- 1.5 Theoretical Properties.- 1.6 Applications.- 1.7 Closure.- References.- 2. Computational Galerkin Methods.- 2.1 Limitations of the Traditional Galerkin Method.- 2.2 Solution for Nodal Unknowns.- 2.3 Use of Low-order Test and Trial Functions.- 2.4 Use of Finite Elements to Handle Complex Geometry.- 2.5 Use of Orthogonal Test and Trial Functions.- 2.6 Evaluation of Nonlinear Terms in Physical Space.- 2.7 Advantages of Computational Galerkin Methods.- 2.8 Closure.- References.- 3. Galerkin Finite-Element Methods.- 3.1 Trial Functions and Finite Elements.- 3.2 Examples.- 3.3 Connection with Finite-Difference Formulae.- 3.4 Theoretical Properties.- 3.5 Applications.- 3.6 Closure.- References.- 4. Advanced Galerkin Finite-Element Techniques.- 4.1 Time Splitting.- 4.2 Least-squares Residual Fitting.- 4.3 Special Trial Functions.- 4.4 Integral Equations.- 4.5 Closure.- References.- 5. Spectral Methods.- 5.1 Choice of Trial Functions.- 5.2 Examples.- 5.3 Techniques for Improved Efficiency.- 5.4 Alternative Spectral Methods.- 5.5 Orthonormal Method of Integral Relations.- 5.6 Applications.- 5.7 Closure.- References.- 6. Comparison of Finite Difference, Finite Element and Spectral Methods.- 6.1 Problems and Partial Differential Equations.- 6.2 Boundary Conditions and Complex Boundary Geometry.- 6.3 Computational Efficiency.- 6.4 Ease of Coding and Flexibility.- 6.5 Test Cases.- 6.6 Closure.- References.- 7. Generalized Galerkin Methods.- 7.1 A Motivation.- 7.2 Theoretical Background.- 7.3 Steady Convection-Diffusion Problems.- 7.4 Parabolic Problems.- 7.5 Hyperbolic Problems.- 7.6 Closure.- References.- Appendix 1 Program BURG1.- Appendix 2 Program BURG4.