Uniformly convergent difference scheme for a singularly perturbed problem of mixed parabolic-elliptic type

One dimensional singularly perturbed problem with discontinuous coefficients is considered. The domain is partitioned into two subdomains and in one of them the we have parabolic reaction-diffusion problem and in the other one elliptic convection-diffusion-reaction equation. The problem is discretized using an inverse-monotone finite volume method on Shishkin meshes. We established almost second-order in space variable global pointwise convergence that is uniform with respect to the perturbation parameter. Numerical experiments support the theoretical results.