A weak Galerkin finite element scheme with boundary continuity for second-order elliptic problems

TL;DR

This paper introduces a flexible weak Galerkin finite element method with boundary continuity for second-order elliptic problems, achieving optimal error estimates and demonstrating robustness through numerical experiments.

Contribution

It presents a novel WG finite element scheme that uses boundary continuity polynomials, optimizing degrees of freedom while maintaining accuracy.

Findings

Optimal error estimates in discrete H^1 and L^2 norms.

Numerical experiments confirm robustness and accuracy.

Flexible polynomial space combinations enhance computational efficiency.

Abstract

A new weak Galerkin (WG) finite element method for solving the second-order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed. The WG method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces. Different combination gives rise to different weak Galerkin finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme, yet without losing the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^1$ 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.