Discrete Besov framework for finite volume approximation of the $p$-Laplacian on non-uniform Cartesian grids

International audienceThis work addresses the problem of a priori error analysis for the finite volume approximation of solutions of the p-laplacian on cartesian grids. We first recall the way we constructed such schemes and the different error bounds we proved in our previous works. Then we concentrate particularly on the case where the exact solution has only weak regularity properties (which are natural for this problem) of Besov kind with derivation index in between 1 and 2. In this framework, the usual techniques to obtain error estimates for finite volumes schemes are not straightfoward to apply. Hence, we propose to take advantage of the variational structure of the equation and the schemes in order to obtain the error bounds. In the case of uniform meshes, this strategy was succesfully applied in a previous work of the authors to obtain optimal estimates. We propose in this work to extend this approach to a more general family of quasi-uniform meshes