Convergence of the Difference Solutions of a Dirichlet Problem With a Discontinuous Derivative of the Boundary Function for a Singularly Perturbed Convection-Diffusion Equation

Abstract

We consider a Dirichlet problem for a singularly perturbed convection-diffusion equation with constant coefficients in a rectangular domain in the case when the convection is parallel to the horizontal faces of the rectangular and directed to the right while the first derivative of the boundary function is discontinuous on the left face. Under these conditions the solution of the problem has a regular boundary layer in the neighborhood of the right face, two characteristic boundary layers near the top and bottom faces, and a horizontal interior layer due to the non-smoothness of the boundary function. We show that on the piecewise uniform Shishkin meshes refined near the regular and characteristic layers, the solution given by the classical five-point upwind difference scheme converges uniformly to the solution of the original problem with almost first-order rate in the discrete maximum norm. This is the same rate as in the case of a smooth boundary function. The numerical results presented support the theoretical estimate. They show also that in the case of the problem with a dominating interior layer the piecewise uniform Shishkin mesh refined near the layer decreases the error and gives the first-order convergence.The article is published in the author’s wording.

Keywords

• convection-diffusion
• singular perturbation
• interior layer
• uniform convergence