Abstract: 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 $\mathcal{O}(∆x^{2/(19+d)})$, where $d$ is the spatial dimension and $∆x$ 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 [24], in which the one-dimensional case was examined using the Kružkov-Carrillo entropy framework.