Error expansion for a first‐order upwind difference scheme applied to a model convection–diffusion problem

A singularly perturbed convection–diffusion problem is considered. The problem is discretized using a simple first‐order upwind difference scheme on general meshes. We derive an expansion of the error of the scheme that enables uniform error bounds with respect to the perturbation parameter in the discrete maximum norm for both a defect correction method and the Richardson extrapolation technique. This generalizes and simplifies results obtained in earlier publications by Fröhner et al.(2001, Numer. Algorithms, 26, 281–299) and by Natividad & Stynes (2003, Appl. Numer. Math., 45, 315–329). Numerical experiments complement our theoretical results.