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.

翻译：我们针对二维奇异摄动对流扩散偏微分方程的数值解提出一种有限差分格式，其解呈现边界层与内部层相互作用，后者由源项间断引起。问题定义在单位正方形域上。二阶导数乘以奇异摄动参数$\epsilon$，而一阶导数项的性质使得流动方向与边界平行。这两个事实意味着解往往呈现指数型与特征型两类层。我们采用一种针对间断特性专门适配的有限差分方法，基于分段均匀（Shishkin）网格求解该问题。我们证明：计算解以独立于摄动参数的速率收敛至真解，且收敛阶接近一阶。我们呈现的数值结果验证了该结论的精确性。