  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
$\mathcal{O}(\Delta x^{2/(19+d)})$, where $d$ is the spatial dimension and
$\Delta x$ 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.
