An upwind difference scheme on a novel Shishkin-type mesh for a linear convection–diffusion problem

We consider an upwind finite difference scheme on a novel layer-adapted mesh (a modification of Shishkinʹs piecewise uniform mesh) for a model singularly perturbed convection–diffusion problem in two dimensions. We prove that the upwind scheme on the modified Shishkin mesh is first-order convergent in the discrete L∞ norm, independently of the diffusion parameter ε, provided only that the perturbation parameter satisfies ε⩽N−1, where O(N2) mesh points are used. The new mesh yields more accurate results than simple upwinding on a standard Shishkin mesh, even though it requires essentially the same computational effort. Numerical experiments support these theoretical results.