Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes

Abstract

In this paper, we derive robust a posteriori error estimates for conforming approximations to a singularly perturbed reaction-diffusion problem on anisotropic meshes, since the solution in general exhibits anisotropic features, e.g., strong boundary or interior layers. Based on the anisotropy of the mesh elements, we improve the a posteriori error estimates developed by Cheddadi et al., which are reliable and efficient on isotropic meshes but fail on anisotropic ones. Without the assumption that the mesh is shape-regular, the resulting mesh-dependent error estimator is shown to be reliable, efficient and robust with respect to the reaction coefficient, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming one, like the piecewise linear finite element one. Our estimates are based on the usual H(div)-conforming, locally conservative flux reconstruction in the lowest-order Raviart-Thomas space on a dual mesh associated with the original anisotropic simplex one. Numerical experiments in 2D confirm that our estimates are reliable, efficient and robust on anisotropic meshes.