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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