Weak Galerkin finite element method for Poisson's equation on polytopal meshes with arbitrary small edges or faces

TL;DR

This paper analyzes a weak Galerkin finite element method for solving Poisson's equation on complex polygonal and polyhedral meshes, establishing optimal error estimates even with very small edges or faces.

Contribution

It introduces and analyzes a weak Galerkin method applicable to meshes with arbitrarily small edges or faces, providing optimal convergence results.

Findings

Optimal convergence order for $H^1$ and $L_2$ errors

Element and edge based error estimates proved

Method applicable to complex polygonal/polyhedral meshes

Abstract

In this paper, the weak Galerkin finite element method for second order elliptic problems employing polygonal or polyhedral meshes with arbitrary small edges or faces was analyzed. With the shape regular assumptions, optimal convergence order for $H^1$ and $L_2$ error estimates were obtained. Also element based and edge based error estimates were proved.