Effective Implementation of the Weak Galerkin Finite Element Methods for the Biharmonic Equation

TL;DR

This paper presents an efficient implementation algorithm for weak Galerkin finite element methods applied to the biharmonic equation, significantly reducing computational complexity while maintaining accuracy.

Contribution

It develops a novel algorithm that eliminates local unknowns, producing a smaller, equivalent global system based on Schur complements for the WG method.

Findings

Reduced global system size by eliminating local unknowns

Proved equivalence between WG method and its Schur complement

Numerical results confirm the efficiency of the implementation

Abstract

The weak Galerkin (WG) methods have been introduced in the references [11, 16] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique.