# Preprint 2015-020

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

## Kenneth Hvistendahl Karlsen, Nils Henrik Risebro and Erlend Briseid Storrøsten

**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.