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

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 & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50, 2729–2743), 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 method, we extend the approach of Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal., 50, 2729–2743) 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.