Lars Wahlbin Abstract It is proposed to work on several problems in the numerical analysis of finite element methods for partial differential and integral equations. A. Maximum-norm estimates for finite element methods in second order hyperbolic problems: In contrast to the situation for elliptic and parabolic problems, the theory in the case of hyperbolic problems is very incomplete. B. Superconvergence in finite element methods: The main problem is to ascertain when (and when not) superconvergence holds up to boundaries. C. Finite element methods in problems with anisotropic diffusion: In the singularly perturbed case, very little is known. D. Finite element methods in partial integro-differential equations: The main problem lies with enormous memory and work requirements in problems with an integral-type memory term. It has been estimated that 2/3 of all computer runs involving approximation of scientific or engineering models are useless: not because they are necessarily wrong but because you do not know that they are right and so cannot trust them. This is particularly annoying when you are formulating a new model and want to find its predictions to test against reality. You do not know whether a peculiar prediction is inherent in the model or crept in via a faulty computer simulation of it. The present proposal is aimed at investigating how a selection of widely used simulation methods behave on mathematical problems with various quirks. It will thus give guidelines to people using numerical methods and help them in judging whether a particular series of computer runs is reliable or not in representing a scientific model.
