A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions

An elliptic system of $M (\ge 2)$ singularly perturbed linear reaction-diffusion equations, coupled through their zero-order terms, is considered on the unit square. This system does not in general satisfy a maximum principle. It is solved numerically using a standard difference scheme on tensor-product Bakhvalov and Shishkin meshes. An error analysis for these numerical methods shows that one obtains nodal $O(N^{-2})$ convergence on the Bakhvalov mesh and $O(N^{-2}\ln^2 N)$ convergence on the Shishkin mesh, where $N$ mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter. The analysis is much simpler than previous analyses of similar problems, even in the case of a single reaction-diffusion equation, as it does not require the construction of an elaborate decomposition of the solution. Numerical results are presented to confirm our theoretical error estimates.