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

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) meth-ods 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 hyper-bolic 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 hy-perbolic 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 Peclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adap-tive 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 re-gion, 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 depen-dent. Extensive numerical experiments are performed to verify the model's robustness for all the abovementioned regimes.(c) 2023 Elsevier Inc. All rights reserved.