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.

[1]  Shuyu Sun,et al.  A Locally Conservative Finite Element Method Based on Piecewise Constant Enrichment of the Continuous Galerkin Method , 2009, SIAM J. Sci. Comput..

[2]  Guang Lin,et al.  Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity , 2014, J. Comput. Phys..

[3]  T. F. Russell.  Relationships among Some Conservative Discretization Methods , 2000 .

[4]  C. Bahriawati,et al.  Three Matlab Implementations of the Lowest-order Raviart-Thomas Mfem with a Posteriori Error Control , 2005 .

[5]  Chunmei Wang,et al.  A Hybridized Weak Galerkin Finite Element Method for the Biharmonic Equation , 2014 .

[6]  Susanne C. Brenner,et al.  A weakly over-penalized symmetric interior penalty method for the biharmonic problem. , 2010 .

[7]  Lin Mu,et al.  L2 error estimation for DGFEM for elliptic problems with low regularity , 2012, Appl. Math. Lett..

[8]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[9]  B. Rivière,et al.  Superconvergence and H(div) projection for discontinuous Galerkin methods , 2003 .

[10]  Junping Wang,et al.  A computational study of the weak Galerkin method for second-order elliptic equations , 2011, Numerical Algorithms.

[11]  Haiying Wang,et al.  Locally Conservative Fluxes for the Continuous Galerkin Method , 2007, SIAM J. Numer. Anal..

[12]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[13]  Jiangguo Liu,et al.  A NOTE ON THE APPROXIMATION PROPERTIES OF THE LOCALLY DIVERGENCE-FREE FINITE ELEMENTS , 2007 .

[14]  B. Rivière,et al.  Estimation of penalty parameters for symmetric interior penalty Galerkin methods , 2007 .

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[16]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[17]  Long Chen,et al.  A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems , 2014, J. Sci. Comput..

[18]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[19]  Junping Wang,et al.  A weak Galerkin finite element method for second-order elliptic problems , 2011, J. Comput. Appl. Math..