A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods

This paper presents a comparative study on the newly introduced weak Galerkin finite element methods (WGFEMs) with the widely accepted discontinuous Galerkin finite element methods (DGFEMs) and the classical mixed finite element methods (MFEMs) for solving second-order elliptic boundary value problems. We examine the differences, similarities, and connection among these methods in scheme formulations, implementation strategies, accuracy, and computational cost. The comparison and numerical experiments demonstrate that WGFEMs are viable alternatives to MFEMs and hold some advantages over DGFEMs, due to their properties of local conservation, normal flux continuity, no need for penalty factor, and definiteness of discrete linear systems.