A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions 1

Let u 2 H be the exact solution of a given self{adjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable nite element space. EEcient and reliable a posteriori estimates of the error j j u ? ~ u j j, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive reenement of the underlying mesh. A well{known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations .