Robust estimates in balanced norms for singularly perturbed reaction diffusion equations using graded meshes

GABRIELA ARMENTANO; ARIEL LUIS LOMBARDI; CECILIA PENESSI

JOURNAL OF SCIENTIFIC COMPUTING

SPRINGER/PLENUM PUBLISHERS

Lugar: New York; Año: 2023

0885-7474

The goal of this paper is to provide almost robust approximations of singularly perturbed reaction-diffusion equations in two dimensions by using finite elements ongraded meshes. When the mesh grading parameter is appropriately chosen, we obtain quasi-optimal 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 super-closeness result, namely, thatthe 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.