A Dual Finite Element Method for a Singularly Perturbed Reaction-Diffusion Problem

Abstract

We present a dual finite element method for a singularly perturbed reaction-diffusion problem. It can be considered a reduced version of the mixed finite element method for approximate solutions. The new method only approximates the dual variables without approximating the primary variable. An approximation for the primary variable is recovered through a simple local $L_2$ projection. Optimal error estimates for the primary and flux variables are obtained. Our method provides a competitive alternative to other existing numerical methods. For example, our approximate solution for the primary variable does not show a significant numerical oscillation, which is observed in the standard Galerkin methods, and we present a confirming numerical example.