A Unified Study of Continuous and Discontinuous Galerkin Methods

TL;DR

This paper provides a comprehensive unified framework for various finite element methods, including conforming, nonconforming, mixed, hybrid, discontinuous Galerkin, hybrid DG, and weak Galerkin methods, analyzing their stability and convergence properties.

Contribution

It introduces a unified analysis that demonstrates stability and convergence of multiple FEMs, including new insights into the limits of WG and HDG methods.

Findings

HDG and WG admit uniform inf-sup conditions

WG converges to a mixed method as stabilization vanishes

HDG converges to a primal method in the limit

Abstract

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.