Robust Estimates in Balanced Norms for Singularly Perturbed Reaction Diffusion Equations Using Graded Meshes

The goal of this paper is to provide almost robust approximations of singularly perturbed reaction-diffusion equations in two dimensions by using finite elements on graded meshes. When the mesh grading parameter is appropriately chosen, we obtain quasioptimal error estimations in a balanced norm for piecewise bilinear elements, by using a weighted variational formulation of the problem introduced by N. Madden and M. Stynes, Calcolo 58(2) 2021. We also prove a supercloseness result, namely, that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the weighted balanced norm, is of higher order than the error itself. We finish the work with numerical examples which show the good performance of our approach.