A multiscale a posteriori error estimate

We introduce a hierarchic a posteriori error estimate for singularly perturbed reaction-diffusion problems. The estimator is based on a Petrov-Galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions. We study the equivalence between the a posteriori estimate and the exact error in the energy norm. Moreover, we prove a relationship between the hierarchic estimator and an explicit residual estimator. The approach provides accurate estimates for the boundary layer regions which is confirmed by numerical experiments. (c) 2004 Elsevier B.V. All rights reserved.