Adaptive refinement for convection-diffusion problems based on a defect-correction technique and finite difference method

Abstract

A difference method is presented for singularly perturbed convection-diffusion problems with discretization error estimates of high order (orderp), which hold uniformly in the singular perturbation parametere. The method is based on the use of a defect-correction technique and special adaptively graded and patched meshes, with meshsizes varying betweenh ande 3/2 h whenp=2, whereh is the meshsize, used in the part of the domain where the solution is smooth, ande 3/2 h is the final meshsize in the boundary layer. Similar constructions hold for interior layers. The correction operator is a monotone operator, enabling the estimate of the error of optimal order in maximum norm. The total number of meshpoints used in ad-dimensional problem isO(e −s)h −d+O(h −d), wheres is 1/p or 1/2p, respectively in the case of boundary or interior layer.