Discrete duality finite volume schemes for Leray-Lions type elliptic problems on general 2D meshes

Discrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo & Omnès in for the Laplace equation, are proposed for nonlinear diffusion problems with non homogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of non-conformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general $W^{-1,p'}(\O)$ source term and $W^{1-\frac{1}{p},p}(\pa\O)$ boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient respectively in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in $W^{2,p}(\O)$. Numerical examples are given, including those on locally refined meshes