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

Folkmar Bornemann
Freie Universität Berlin
Arnimalle 2-6
D-14195 Berlin

Bodo Erdmann and Ralf Kornhuber
Konrad-Zuse-Zentrum Berlin
Heilbronner Str. 10
D-10711 Berlin


Let u in H be the exact solution of a given self-adjoint elliptic boundary value problem, which is approximated by some uS in S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error ||u-uS||, measuring the (local) quality of uS, play a crucial role in termination criteria and in the adaptive refinement 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.

Contributed April 8, 1994.