Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give epsilon -uniform maximum norm error estimates O(N-2 ln(2) N(+tau)) and O(N-2(+tau)). respectively. for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, tau is the rime mesh-interval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition.