Mathematical Theory of Finite and Boundary Element Methods

I: An Analysis of the Finite Element Method for Second Order Elliptic Boundary Value Problems.- O. Introduction.- 1. Some function spaces, notation and preliminaries.- 2. Some finite element spaces and their properties.- 3. Orthogonal projections onto finite element spaces in L2, in H1 and H01.- 4. Galerkin finite element method for second order elliptic boundary value problems. Basic Hl and L2 estimates.- 5. Indefinite second order elliptic problems.- 6. Local error estimates.- 7. An introduction to grid refinement. An application to boundary value problems with non-convex corners.- 8. Maximum norm estimates for the L2 projection. A method using weighted norms.- 9. Maximum norm estimates for the Galerkin finite element method for second order elliptic problems.- References.- II: The Finite Element Method for Parabolic Problems.- 1. Introduction.- 2. Non-smooth data error estimates for the semidiscrete problem.- 3. Completely discrete schemes.- 4. A nonlinear problem.- References.- III: Boundary Element Methods for Elliptic Problems.- 1 Boundary Integral Equations.- 1.1 The exterior Neumann problem for the Laplacian.- 1.2 Exterior viscous flow problems.- 1.3 Scattering problems in acoustics.- 1.4 Some problems of elastostatics.- 1.5 The boundary integral equations of the direct approach for general elliptic boundary value problems of even order.- 2 The Characterization of Boundary Integral Operators and Galerkin Boundary Element Methods.- 2.1 The representation and the order of boundary integral operators.- 2.2 Variational formulation and strong ellipticity.- 2.3 Boundary element Galerkin methods.- 3 Collocation Methods.- 3.1 Collocation with smoothest splines of piecewise odd polynomials.- 3.2 Naive spline collocation for n = 2 on almost uniform partitions.- 4 Concluding Remarks.