Error in the upwind differencing of the convection-diffusion equation

In convection-diffusion problems a first-order upwind difference approximation is usually used on a locally refined mesh to get a wiggle free solution, which is then corrected using some form of deferred correction to get second-order accuracy. Pereyra (1966) has shown that improving accuracy, of a low-order scheme using deferred correction, requires the existence of an asymptotic expansion for the discretization error of the scheme concerned. In this paper bounds on the discrezazation error are obtained when the first-order upwind difference scheme is applied to the convection-diffusion equation on a locally refined mesh, both in one and two dimensions. Existence of an asymptotic expansion for the discretization error is established. Different formulae, at the nodes separating subregions of differing mesh sizes, are analysed and numerical tests performed to illustrate validity of the resulting expressions.