Uniform Grid Approximation of Nonsmooth Solutions of a Singularly Perturbed Convection - Diffusion Equation with Characteristic Layers

Abstract

A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients is considered in a square with the Dirichlet conditions imposed on the two sides, which are orthogonal to the flow direction, and with the Neumann conditions on the other two sides. Sufficient smoothness of the right-hand side and that of the boundary functions is assumed, which ensures the required smoothness of the solution in the considered domain, except the neighborhoods of the corner points. At the corner points themselves, zero order compatibility conditions alone are assumed to be satisfied.

For the numerical solution to the posed problem a nonuniform monotonous difference scheme is used on a rectangular piecewise uniform Shishkin grid. Non-uniformity of the scheme means that the form of the difference equations, which are used for the approximation, is not the same in different grid points but it depends on the value of the perturbing parameter.

Under assumptions made a uniform convergence with respect to ε of the numerical solution to the precise solution is proved in a discrete uniform metric at the rate O(N ^− 3/2ln ² N), where N is the number of the grid points in each coordinate direction.