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.}