Discontinuous Galerkin Methods Based on Weighted Interior Penalties for Second Order PDEs with Non-smooth Coefficients

摘要

We develop and analyze a Discontinuous Galerkin (DG) method based on weighted interior penalties (WIP) applied to second order (elliptic) PDEs and in particular to advection-diffusion-reaction equations featuring non-smooth and possibly vanishing dif-fusivity. First of all, looking at the derivation of a DG scheme with a bias to domain decomposition methods, we carefully discuss the set up of the discretization scheme in a general framework putting into evidence the helpful role of the weights and the connection with the well known Local Discontinuous Galerkin schemes (LDG). Then, we address the a-priori error analysis of the method, recovering optimal error estimates in suitable norms. By virtue of the introduction of the weighted penalties, these results turn out to be robust with respect to the diffusion parameter. Furthermore, we discuss the derivation of an a-posteriori local error indicator suitable for advection-diffusion-reaction problems with highly variable, locally small diffusivity. All the theoretical results are illustrated and discussed by means of numerical experiments.