Optimal Rate of Convergence for Anisotropic Vanishing Viscosity Limit of a Scalar Balance Law

An open question in numerical analysis of multidimensional scalar conservation laws discretized on non-structured grids is the optimal rate of convergence. The main difficulty relies on a priori BV bounds which cannot be derived by opposition to the case of structured (cartesian) grids. In this paper we consider a related question for a corresponding continuous model, namely the vanishing viscosity method for a multidimensional scalar conservation law with a general diffusion matrix which is only bounded. Then, BV estimates are not available here and we prove the h1/2 convergence rate. Our strategy of proof differs from the classical method of Kuznetzov. It consists in using in an accurate way the entropy dissipation due to the parabolic terms. The dissipation of the conservation law is not strong enough and we thus consider an auxiliary parabolic problem to compensate that. Using the kinetic formulation and the related uniqueness method also helps to avoid unessential technicalities.