Computational Galerkin 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.