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

摘要

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