Parameter-uniform numerical method for a two-dimensional singularly perturbed convection–reaction–diffusion problem with interior and boundary layers

We consider a two-dimensional singularly perturbed convection–reaction–diffusion problem that has discontinuities, along lines parallel to x- and y-axes, in the source term, as well as in the convection and reaction coefficients. The coefficient of the highest-order term is a small positive parameter denoted by ε . Due to the discontinuities, the solution exhibits layers in the interior of the domain, in addition to boundary layers. We propose a decomposition of the solution that yields sharp bounds on its derivatives. A finite difference scheme is constructed on an appropriate Shishkin mesh, and it is established that the computed solution is almost first-order, parameter-uniformly convergent. Numerical results are given to support the theoretical results.