Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients

We consider a singularly perturbed elliptic problem in two dimensions with discontinuous coefficients and line interface. The second-order derivatives are multiplied by a small parameter ε2. The solution of this problem exhibits boundary, interior and corner layers. A finite volume difference scheme is constructed on partially uniform layer-adapted (Shishkin mesh). We prove that it yields an accurate approximation of the solution both inside and outside these layers. Error estimates in the discrete maximum norm that hold true uniformly in the perturbation parameter ε are obtained. Numerical experiments that agree with the theoretical result are given.