Generalized Weak Galerkin Finite Element Methods for Biharmonic Equations

TL;DR

This paper introduces a generalized weak Galerkin finite element method for solving biharmonic equations, allowing flexible polynomial functions on complex geometries, with proven error estimates and demonstrated accuracy.

Contribution

The paper presents a novel generalized weak Galerkin scheme with a new discrete second order derivative, enabling arbitrary polynomial combinations on polygonal/polyhedral meshes.

Findings

Error estimates in discrete H^2 and L^2 norms established.

Numerical results confirm the method's accuracy and flexibility.

Applicable to complex polygonal/polyhedral meshes.

Abstract

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H^2 norm and a L^2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.