A posteriori error estimates for elliptic problems in two and three space dimensions

Let u is an element of H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some (u) over tilde is an element of S, S being a suitable finite-element space. Efficient and reliable a posteriori estimates of the error \\u - (u) over tilde\\, measuring the (local) quality of (u) over tilde, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wed-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this 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.