A Hybridized Formulation for the Weak Galerkin Mixed Finite Element Method

TL;DR

This paper introduces a hybridized formulation for the weak Galerkin mixed finite element method, reducing computational complexity and providing optimal error estimates for second order elliptic equations.

Contribution

It develops a hybridized version of WG-MFEM that simplifies the linear system and achieves superconvergence, with theoretical error bounds and numerical validation.

Findings

Reduced linear system involving only Lagrange multipliers

Optimal-order error estimates for hybridized WG-MFEM

Numerical results confirm superconvergence and theoretical predictions

Abstract

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. Some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier.