A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

TL;DR

This paper introduces a new stabilizer-free weak Galerkin finite element method on polytopal meshes that achieves superconvergence with convergence rates two orders higher than traditional methods, validated through numerical experiments.

Contribution

It develops a novel stabilizer-free WG method with significantly improved convergence rates on polytopal meshes, surpassing existing methods.

Findings

Achieves superconvergence with two orders higher than optimal.

Demonstrates effectiveness in 2D and 3D numerical tests.

Applicable to low and high order elements.

Abstract

A weak Galerkin (WG) finite element method without stabilizers was introduced in [J. Comput. Appl. Math., 371 (2020). arXiv:1906.06634] on polytopal mesh. Then it was improved in [arXiv:2008.13631] with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the $L^2$ norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.