Singularly perturbed reaction-diffusion
problems exhibit in general solutions with anisotropic features,
e.g. strong boundary and/or interior layers.
This anisotropy is reflected in
discretization by using meshes
with anisotropic elements. The quality of the numerical solution
rests on the robustness of the a posteriori error estimator with
respect to both, the perturbation parameters of the problem
and the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested by
Ainsworth and Babuška has been proved to be robust for the case of singular perturbation.
In the present work we investigate the modified method on anisotropic meshes.
The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent to
the equilibrated residual method in the case of isotropic meshes
and is proved to be robust on anisotropic meshes as well.
Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element.
In practical computations an approximate solution to this local problem was successfully computed.
Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This
demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the
method is not affected by the approximate solution of the local problem.