AbstractThe paper describes a numerical scheme for solving a convection–diffusion elliptic system with very small diffusion coefficients. This iterative numerical procedure is unconditionally stable and converges very rapidly. Although only linear equations are considered here, this technique can be easily extended to non‐linear equations, while keeping its main features as for the linear case. The numerical experiments presented are quite general and confirm most of these features. These examples also show a good way of implementing this sch