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.

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

“…One may be able to construct, for example, a weakly over-penalized interior penalty (IP) method (like [8]) or an interior penalty discontinuous Galerkin (IPDG) method with optimal convergence rate robust with respect to the penalization parameter [61] with piecewise cubic polynomials. More discussions may be carried out by the aid of the framework of [28].…”

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

“…From [3,22], we set in (1) and g 2 � e − (1/2)(y− 1/2) 2 in (21). Similarly, we define the numerical fluxes for the rectangular waveguide ( 1) and ( 20) and ( 22) as the following:…”

“…It is also well known that the discontinuous Galerkin methods are flexible and highly parallelizable, and hence discontinuous Galerkin methods are widely used to solve the Helmholtz equation numerically, such as interior penalty discontinuous Galerkin method [15], hybridizable discontinuous Galerkin method [16,17], local discontinuous Galerkin method [18], and the references therein. However, the local discontinuous Galerkin method [19][20][21] is known to be more physical and flexible on designing discontinuous Galerkin schemes. Two local discontinuous Galerkin methods are studied in [18], where the P − 1 1 − P − 1 1 finite element pair was used to approximate the flux variable and the scattered field.…”

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

“…In recent years, DG methods have been developed to solve various problems, and some unified analysis for a finite element including DG methods has recently been presented in the works of Hong et al In this section, as motivated by the works of Schötzau et al and Hong et al, we propose discretizations of the MPET model problem (11). These discretizations preserve the divergence condition (namely Equation ) pointwise, which results in a strong conservation of mass (see Proposition ).…”