We analyze families of primal high-order hybridizable discontinuous Galerkin (HDG) methods for solving degenerate (second-order) elliptic problems. One major trouble regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptic in the rest. This problem is thus quite delicate to solve since the exact solution is discontinuous at interfaces separating both elliptic and hyperbolic parts. The proposed HDG method is developed in a unified and compact fashion. It can efficiently handle pure diffusive or advective regimes and intermediate regimes that combine the above mechanisms for a wide range of P\'eclet numbers, including the delicate situation of local evanescent diffusion. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is then considered. A $\theta$-upwind-based scheme is favored for the hyperbolic region, and an inspired Scharfetter--Gummel-based technique is preferred for the elliptic region. The well-posedness of the HDG method is also discussed by analyzing the consistency and discrete coercivity properties. Extensive numerical experiments are finally considered to verify the model's robustness for all the abovementioned regimes.

翻译：我们分析原始高序混合不连续的Galerkin (HDG) 家庭, 以解决退化( 第二顺序) 椭圆形问题。 有关这一类PDE 的主要难题之一是其数学性质, 其数学性质可能不统一。 由于扩散术语的局部退化, 它在一个分区可以是纯超脱型的, 而在其余的分区可能是椭圆形的。 这个问题因此非常微妙, 因为精确的解决方案在将椭圆形和双曲形部分隔开的界面上是不连贯的。 拟议的HDG 方法是以统一和紧凑的方式开发的。 它能够有效地处理纯的 diffusive 或 advictive 制度和中间制度, 将以上各种机制结合在一起, 其范围可能不统一。 由于扩散术语的局部性, 它可能在一个分区中纯粹是双曲形的, 而在其它的分区中, 它可能是一个纯粹的双向的, 。 因此, 将一个基于跳形和双曲线部分的适应稳定战略 。 一个基于 计划被偏爱极区域,, 并且 受启发的Scharf- Gumel- gemel- gemel- sy- sy- sy 技术 也被最后被讨论到 Qalallivicolvical liver liver liveralalalalal liviolvical 。