A Hybridized Weak Galerkin Finite Element Method for the Biharmonic Equation

TL;DR

This paper introduces a hybridized weak Galerkin finite element method for the biharmonic equation, utilizing a Lagrange multiplier for boundary derivatives, with proven optimal error estimates and an efficient computational algorithm.

Contribution

It develops a novel hybridized weak Galerkin scheme with a Lagrange multiplier, providing optimal error estimates and a reduced computational system.

Findings

Optimal order error estimates established

Efficient Schur complement algorithm derived

Numerical approximation of derivatives verified

Abstract

This paper presents a hybridized formulation for the weak Galerkin finite element method for the biharmonic equation. The hybridized weak Galerkin scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier is verified to provide a numerical approximation for certain derivatives of the exact solution. An optimal order error estimate is established for the numerical approximations arising from the hybridized weak Galerkin finite element method. The paper also derives a computational algorithm (Schur complement) by eliminating all the unknown variables on each element, yielding a significantly reduced system of linear equations for unknowns on the boundary of each element.