[Paper Review] The Basics of Weak Galerkin Finite Element Methods

Junping Wang, Xiu Ye|arXiv (Cornell University)|Jan 28, 2019

Advanced Numerical Methods in Computational Mathematics33 references3 citations

TL;DR

This paper clarifies the conceptual and methodological distinctions between the weak Galerkin finite element method (WG-FEM) and the hybridizable discontinuous Galerkin (HDG) method, demonstrating that while both share some mathematical roots, they are based on different philosophical foundations. The authors prove that the primal WG-FEM is not equivalent to existing HDG methods, even when using the same finite element spaces, through theoretical analysis and a numerical example showing a fundamental mismatch in their variational formulations.

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].

Motivation & Objective

• To correct misconceptions in [6] about the equivalence between weak Galerkin and HDG methods.
• To clarify the philosophical and methodological distinctions between WG-FEM and HDG, despite shared mathematical foundations.
• To demonstrate, both theoretically and via an example, that the primal WG-FEM is not equivalent to existing HDG schemes.
• To provide a systematic comparison of hybridized mixed WG-FEM and HDG formulations using a common basis function framework.
• To challenge the claim that WG-FEM is merely a rewriting of HDG, emphasizing the distinct design principles of each method.

Proposed method

• The authors develop three variational forms for the model PDE: primal, primal-mixed, and mixed (dual-mixed), serving as the foundation for both WG and HDG methods.
• For each variational form, they construct corresponding weak Galerkin FEMs: primal WG-FEM, primal-mixed WG-FEM, and mixed WG-FEM, using discrete weak gradients and stabilizers.
• They derive a hybridized formulation of the mixed WG-FEM by applying the Fraeijs de Veubeke method, analogous to hybridized mixed FEM.
• The hybridized mixed WG-FEM is reformulated to allow direct comparison with HDG, using identical finite element spaces and basis functions.
• The comparison is conducted by aligning the unknowns and governing equations of both methods under the same discretization framework.
• A counterexample is constructed using a specific coefficient function a(x) = 1 + x, showing that the bilinear forms in the two methods do not match, proving non-equivalence.

Experimental results

Research questions

• RQ1 Is the primal weak Galerkin finite element method equivalent to existing hybridizable discontinuous Galerkin methods when using the same finite element spaces and basis functions?
• RQ2 What are the fundamental philosophical and methodological differences between the weak Galerkin and HDG methods, despite shared mathematical roots?
• RQ3 Can the hybridized mixed WG-FEM be reformulated to allow a direct, fair comparison with the HDG method in terms of unknowns and linear systems?
• RQ4 Does the second formulation of HDG introduced in [6] represent a true reformulation of the primal WG-FEM, and if so, under what conditions?
• RQ5 Is the claim that '2013 WG methods are mixed methods' accurate, particularly for problems with variable coefficients?

Key findings

• The primal WG-FEM is not equivalent to existing HDG methods, even when using the same finite element spaces and basis functions, as shown by a theoretical counterexample.
• The bilinear forms in the weak Galerkin and HDG methods differ fundamentally: for a specific test case with a(x) = 1 + x, the identity (9.11) fails because 3/2 (ln 2)² ≠ ln 2.
• The second formulation of HDG introduced in [6] is shown to be a reformulation of the primal WG-FEM under specific choices of finite element spaces and stabilizer parameters.
• The authors reject the claim that WG-FEM is merely a rewriting of HDG, arguing that WG-FEM is a distinct numerical methodology based on unique foundational principles.
• The statement in [6] that the 2013 WG methods are mixed methods is incorrect, especially for problems with variable diffusion coefficients a(x).
• The hybridized mixed WG-FEM and HDG share structural similarities but differ in their underlying formulations and solution mechanisms, confirming they are not the same method.