We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local L 1-error between the exact and numerical solutions is O (Δx 2/(19+ d)), where d is the spatial dimension and Δx is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation. This paper is a continuation of [KH Karlsen, NH Risebro EB Storrøsten, Math. Comput. 83 (2014) 2717–2762], in which the one-dimensional case was examined using the Kruzkov− Carrillo entropy framework.