  We propose a finite difference scheme for the numerical solution of a
two-dimensional singularly perturbed convection-diffusion partial differential
equation whose solution features interacting boundary and interior layers, the
latter due to discontinuities in source term. The problem is posed on the unit
square. The second derivative is multiplied by a singular perturbation
parameter, $\epsilon$, while the nature of the first derivative term is such
that flow is aligned with a boundary. These two facts mean that solutions tend
to exhibit layers of both exponential and characteristic type. We solve the
problem using a finite difference method, specially adapted to the
discontinuities, and applied on a piecewise-uniform (Shishkin). We prove that
that the computed solution converges to the true one at a rate that is
independent of the perturbation parameter, and is nearly first-order. We
present numerical results that verify that these results are sharp.
