A Posteriori Error Estimates for Elliptic Problems 
      in Two and Three Space Dimensions
 
           Folkmar Bornemann
Freie Universit\"at Berlin, Arnimalle 2--6, D-14195 Berlin

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

Abstract:
Let u in H be the exact solution of a given self--adjoint elliptic boundary
value problem, which is approximated by some u_S in S, S being a suitable
finite element space.  Efficient and reliable a posteriori estimates of the
error || u - u_S ||, measuring the (local) quality of u_S, 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.