Stabilizer-free Weak Galerkin Methods for Quad-Curl Problems on polyhedral Meshes without Convexity Assumptions
Chunmei Wang, Shangyou Zhang
TL;DR
This work develops a stabilizer-free weak Galerkin method for 3D quad-curl problems on polyhedral meshes, including non-convex elements, by employing bubble functions to eliminate stabilization terms. The proposed scheme preserves a symmetric, positive definite system, uses discrete weak gradient and curl-curl operators, and achieves optimal discrete-norm error with optimal-order $L^2$ error for $k>2$ (suboptimal for $k=2$). A complete theoretical framework is established, including well-posedness, error equations, energy and $L^2$ error estimates, and duality-based analyses, complemented by extensive numerical tests on convex and non-convex meshes. The method demonstrates robustness, efficiency, and accuracy in both the theoretical and practical settings, extending stabilizer-free WG methods to high-order 3D quad-curl problems on general polyhedral meshes.
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.