A Discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity

We propose and analyze a symmetric weighted interior penalty (SWIP) method to approximate in a Discontinuous Galerkin framework advection-diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high P´eclet numbers) on tted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh-size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh-size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell P´eclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.