A difference scheme of improved accuracy for a quasilinear singularly perturbed elliptic convection-diffusion equation.

A Dirichlet boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion equation on a strip is considered. For such a problem, a difference scheme based on classical approximations of the problem on piecewise uniform meshes condensing in the layer converges epsilon-uniformly with an order of accuracy not more than 1. We construct a linearized iterative scheme based on the nonlinear Richardson scheme, where the nonlinear term is computed using the sought function taken at the previous iteration; the solution of the iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of a geometry progression epsilon-uniformly with respect to the number of iterations. The use of lower and upper solutions of the linearized iterative Richardson scheme as a stopping criterion allows us during the computational process to define a current iteration under which the same epsilon-uniform accuracy of the solution is achieved as for the nonlinear Richardson scheme