Stabilizer-free Weak Galerkin Methods for Quad-Curl Problems on polyhedral Meshes without Convexity Assumptions

TL;DR

This paper develops a stabilizer-free weak Galerkin finite element method for 3D quad-curl problems on polyhedral meshes, extending applicability to non-convex elements and providing optimal error estimates.

Contribution

It introduces a stabilizer-free WG method that works on non-convex polyhedral meshes without convexity assumptions, a significant improvement over existing methods.

Findings

Achieves optimal error estimates in a discrete norm.

Provides optimal-order $L^2$ error estimates for $k>2$.

Demonstrates efficiency and accuracy through numerical experiments.

Abstract

This paper introduces an efficient stabilizer-free weak Galerkin (WG) finite element method for solving the three-dimensional quad-curl problem. Leveraging bubble functions as a key analytical tool, the method extends the applicability of stabilizer-free WG approaches to non-convex elements in finite element partitions-a notable advancement over existing methods, which are restricted to convex elements. The proposed method maintains a simple, symmetric, and positive definite formulation. It achieves optimal error estimates for the exact solution in a discrete norm, as well as an optimal-order $L^2$ error estimate for $k>2$ and a sub-optimal order for the lowest order case $k=2$. Numerical experiments are presented to validate the method's efficiency and accuracy.