A parameter-robust numerical method for a system of
reaction-diffusion equations in two dimensions
R. Bruce Kellogg, Niall Madden and Martin Stynes
Abstract
A system of $M (\geq 2)$ coupled singularly perturbed linear
reaction-diffusion equations is considered on the unit square.
Under certain hypotheses on the coupling, a maximum principle is
established for the differential operator. The relationship
between compatibility conditions at the corners of the square and
the smoothness of the solution on the closed domain is fully
described. A decomposition of the solution of the system is
constructed. A finite difference method for the solution of the
system on a Shishkin mesh is presented and it is proved that the
computed solution is second-order accurate (up to a logarithmic
factor). Numerical results are given to support this result and to
investigate the effect of weaker compatibility assumptions on the
data.
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