A Unified Study of Continuous and Discontinuous Galerkin Methods
For researchers in numerical analysis, this work provides a theoretical unification of diverse finite element methods, revealing connections and gaps, though it is primarily theoretical and incremental in nature.
This paper presents a unified framework for analyzing various finite element methods, including conforming, nonconforming, mixed, hybrid, DG, HDG, and WG methods, showing that HDG and WG admit uniform inf-sup conditions and that WG converges to a mixed method while HDG converges to a primal method in the limit of stabilization parameters.
A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.