On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

Abstract

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L_∞^h almost with the first order. This theoretical result is confirmed via numerical analysis.