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Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction{diusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic nite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is ecient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem suciently well. A numerical example supports the analysis of our anisotropic error estimator.

Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes