Defect correction on Shishkin-type meshes

We consider a defect-correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for a model singularly perturbed convection–diffusion problem in one dimension on a class of Shishkin-type meshes. The method is shown to be convergent, uniformly in the diffusion parameter , of second order in the discrete maximum norm. As a corollary we derive error bounds for the gradient approximation of the upwind scheme. Numerical experiments support our theoretical results.