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

We consider Discontinuous Galerkin approximations of advection-diffusion equations with anisotropic and discontinuous diffusivity, and propose the symmetric weighted interior penalty (SWIP) method for better coping with locally vanishing diffusivity. The analysis yields convergence results for the natural energy norm that are optimal (with respect to mesh-size) and robust (fully independent of the 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 in the dominant advection regime. In the dominant diffusivity regime, an optimal convergence result for the theL-norm is also recovered. Numerical results are presented to illustrate the performance of the scheme.