A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

TL;DR

This paper introduces a flexible weak Galerkin finite element method using reduced polynomials for biharmonic equations, achieving optimal error estimates and demonstrating high accuracy and efficiency through numerical experiments.

Contribution

It develops a novel WG finite element scheme with reduced polynomial order, optimizing the balance between computational cost and accuracy.

Findings

Achieves optimal error estimates in discrete $H^2$ and $L^2$ norms.

Demonstrates robustness and reliability through numerical experiments.

Reduces the number of unknowns without sacrificing accuracy.

Abstract

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.