Besov regularity and new error estimates for finite volume approximations of the p-laplacian
Résumé
In a previous work, we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in $W^{2,p}$. Actually, $W^{2,p}$ is not a natural space for solutions of the p-laplacian in the case $p>2$. Indeed, for general $L^{p\'}$ data it can be shown that the solution only belongs to the Besov space $B^{1+\\frac{1}{p-1},p}_\\infty$. In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in $L^{p\'}$. We then obtain new error estimates for such solutions in the case of uniform meshes.