The Basics of Weak Galerkin Finite Element Methods
Junping Wang, Xiu Ye
TL;DR
This work investigates the relationship between weak Galerkin FEM (WG-FEM) and HDG, aiming to dispel misconceptions and clarify how WG-FEM fits within a broader variational discretization framework. It succinctly presents the primal, primal-mixed, and dual-mixed variational formulations for a second-order elliptic model problem and derives corresponding conforming FEMs, then develops several WG-based schemes (Primal WG-FEM, Primal-Mixed WG-FEM, and Mixed WG-FEM) with discrete weak gradients and divergences plus stabilizers. The authors then introduce Hybridized Mixed WG-FEM, reformulate it to facilitate comparison with HDG, and demonstrate that HDG is a special case of HM-WG-FEM only under sufficiently rich trace spaces; in general, WG-FEM and HDG are distinct methodologies. The paper concludes with critical remarks on HDG formulations, illustrating non-equivalence in the general case and arguing for WG-FEM as a flexible, philosophy-driven discretization framework with broader applicability and potential advantages in numerical PDEs.
Abstract
The goal of this article is to clarify some misunderstandings and inappropriate claims made in [6] regarding the relation between the weak Galerkin (WG) finite element method and the hybridizable discontinuous Galerkin (HDG). In this paper, the authors offered their understandings and interpretations on the weak Galerkin finite element method by describing the basics of the WG method and how WG can be applied to a model PDE problem in various variational forms. In the authors' view, WG-FEM and HDG methods are based on different philosophies and therefore represent different methodologies in numerical PDEs, though they share something in common in their roots. A theory and an example are given to show that the primal WG-FEM is not equivalent to the existing HDG [9].