A first-order system Petrov-Galerkin discretisation for a reaction-diffusion problem on a fitted mesh

James Adler, Scott MacLachlan, and Niall Madden,

Abstract

We consider the numerical solution, by a Petrov-Galerkin finite-element method, of a singularly perturbed reaction-diffusion differential equation posed on the unit square. In Lin and Stynes (2012), it is argued that the natural energy norm, associated with a standard Galerkin approach, is not an appropriate setting for analysing such problems, and there they propose a method for which the natural norm is ``balanced''. In the style of a first-order system least squares (FOSLS) method, we extend the approach of Lin and Stynes (2012) by introducing a constraint which simplifies the associated finite-element space and the method's analysis. We prove robust convergence in a balanced norm on a piecewise uniform (Shishkin) mesh, and present supporting numerical results. Finally, we demonstrate how the resulting linear systems are solved optimally using multigrid methods.}