Numerical investigation on weak Galerkin finite elements

TL;DR

This paper conducts a comprehensive numerical study on weak Galerkin finite elements, analyzing their stability, convergence, and performance through extensive experiments to guide future research and application.

Contribution

It provides a detailed numerical comparison of various weak Galerkin elements, highlighting their performance differences and opening new research directions.

Findings

Different WG elements exhibit varying stability and convergence properties.

Numerical experiments reveal which WG elements outperform others in specific scenarios.

The study offers a detailed performance guide for selecting WG elements.

Abstract

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. The novel idea of weak Galerkin finite element methods 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 generates different weak Galerkin finite elements. The purpose of this paper is to study stability, convergence and supercloseness of different WG elements by providing many numerical experiments recorded in 31 tables. These tables serve two purposes. First it provides a detail guide of the performance of different WG elements. Second, the information in the tables opens new research territory why some WG elements outperform others.