We analyze a Discontinuous Galerkin method for a problem with linear advection–reaction and [Formula: see text]-type diffusion, with Sobolev indices [Formula: see text]. The discretization of the diffusion term is based on the full gradient including jump liftings and interior-penalty stabilization while, for the advective contribution, we consider a strengthened version of the classical upwind scheme. The developed error estimates track the dependence of the local contributions to the error on the local Péclet numbers. A set of numerical tests support the theoretical derivations.