A Unified Study of Continuous and Discontinuous Galerkin Methods

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.

39 pages

References in corpus (5)

• Parameter-robust stability of classical three-field formulation of Biot's consolidation model
• Nonconforming Finite Element Spaces for $2m$-th Order Partial Differential Equations on ${\mathbb{R}}^n$ Simplicial Grids When $m=n+1$
• ${\mathcal{P}}_m$ Interior Penalty Nonconforming Finite Element Methods for $2m$-th Order PDEs in ${\mathbb{R}}^n$
• Quasi-optimal nonconforming methods for symmetric elliptic problems. III -- DG and other interior penalty methods
• New Hybridized Mixed Methods for Linear Elasticity and Optimal Multilevel Solvers

Cited by in corpus (1)

• An Extended Galerkin analysis in finite element exterior calculus