On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions

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