Equivalence of Weak Galerkin Methods and Virtual Element Methods for Elliptic Equations

TL;DR

This paper demonstrates the equivalence between weak Galerkin methods and virtual element methods for elliptic equations, enabling transfer of ideas and techniques between these numerical approaches.

Contribution

It introduces a modified weak Galerkin method and establishes its equivalence to a new virtual element method, connecting the two frameworks.

Findings

Modified weak Galerkin method is equivalent to a new virtual element method

Original weak Galerkin method is equivalent to non-conforming virtual element method

Connections facilitate transfer of ideas and techniques between methods

Abstract

We propose a modification of the weak Galerkin methods and show its equivalence to a new version of virtual element methods. We also show the original weak Galerkin method is equivalent to the non-conforming virtual element method. As a consequence, ideas and techniques used for one method can be transferred to another. The key of the connection is the degree of freedoms.