These are the lecture notes of the seminar "Mathematische Theorie der finiten Element und Randelementmethoden" organized by the "Deutsche Mathematiker-Vereinigung" and held in Dusseldorf from 07. - 14. of June 1987. Finite element methods and the closely related boundary element methods nowadays belong to the standard routines for the computation of solutions to boundary and initial boundary value problems of partial differential equations with many applications as e.g. in elasticity and thermoelasticity, fluid mechanics, acoustics, electromagnetics, scatter ing and diffusion. These methods also stimulated the development of corresponding mathematical numerical analysis. I was very happy that A. Schatz and V. Thomee generously joined the adventure of the seminar and not only gave stimulating lectures but also spent so much time for personal discussion with all the participants. The seminar as well as these notes consist of three parts: 1. An Analysis of the Finite Element Method for Second Order Elliptic Boundary Value Problems by A. H. Schatz. II. On Finite Elements for Parabolic Problems by V. Thomee. III. I30undary Element Methods for Elliptic Problems by \V. L. Wendland. The prerequisites for reading this book are basic knowledge in partial differential equations (including pseudo-differential operators) and in numerical analysis. It was not our intention to present a comprehensive account of the research in this field, but rather to give an introduction and overview to the three different topics which shed some light on recent research.

Contenu

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.