Families of hybridized interior penalty discontinuous Galerkin methods for locally degenerate advection-diffusion-reaction problems

Abstract

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptical in the rest. This problem is thus quite delicate to solve since the exact solution can be discontinuous at interfaces separating the elliptic and hyperbolic parts. The proposed hybridized interior penalty DG (H-IP) method is developed in a unified and compact fashion. It can handle pure-diffusive or -advective regimes as well as intermediate regimes that combine these two mechanisms for a wide range of Péclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is considered. An upwind-based scheme using a Lax–Friedrichs correction is favored for the hyperbolic region, and a Scharfetter–Gummel-based technique is preferred for the elliptic one. The well-posedness of the H-IP method is also briefly discussed by analyzing the strong consistency and the discrete coercivity condition in a self-adaptive energy-norm that is regime dependent. Extensive numerical experiments are performed to verify the model’s robustness for all the abovementioned regimes.