Computer Methods in Applied Mechanics and Engineering 82 (1990) 301-322 North-holland Adaptive Finite Element Methods for Diffusion and Convection Problems

In this note we give a survey of some recent results on adaptive finite element methods obtained in collaboration with Eriksson, (see [1-8]). As model problems we shall consider the heat equation including the corresponding stationary Poisson equation representing diffusiondominated problems, and also linear convection-dominated convection-diffusion problems. Together, these problems cover the basic linear partial differential equations of parabolic, elliptic and (first order) hyperbolic type. In each of these cases our goal is to solve the following problem (Problem A): Given a norm II. II, a tolerance TOL > 0, and a piecewise polynomial finite element discretization of a certain type (e.g., piecewise polynomials of a certain given degree), design an algorithm for constructing a mesh T with (nearly) minimal number of degrees of freedom, such that