We propose and analyze a symmetric weighted interior penalty (SWIP) method to approximate in a Discontinuous framework advection-diffusion equations with anisotropic and discontinuous diffusivity. originality of the method consists in the use of diffusivity-dependent weighted averages better cope with locally small diffusivity (or equivalently with locally high P ́eclet numbers) on tted. The analysis yields convergence results for the natural energy norm that are optimal with respect mesh-size and robust with respect to diffusivity. The convergence results for the advective derivative optimal with respect to mesh-size and robust for isotropic diffusivity, as well as for anisotropic diffusivity the cell P ́eclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large. Numerical results are presented to illustrate the performance of the proposed scheme.
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Ern et al. (2008) studied this question.