Weak Galerkin Finite Element Methods for Quad-Curl Problems

TL;DR

This paper presents a novel weak Galerkin finite element method for solving three-dimensional quad-curl problems, demonstrating stability, optimal error estimates, and superconvergence through theoretical analysis and numerical experiments.

Contribution

The paper introduces a new WG finite element method for 3D quad-curl problems with proven stability and optimal error estimates, including superconvergence observations.

Findings

Method is stable and accurate with optimal error estimates.

Numerical experiments confirm efficiency and superconvergence.

L2 error estimates are optimal except for lowest orders.

Abstract

This article introduces a weak Galerkin (WG) finite element method for quad-curl problems in three dimensions. It is proved that the proposed WG method is stable and accurate in an optimal order of error estimates for the exact solution in discrete norms. In addition, an $L^2$ error estimate in an optimal order except the lowest orders $k=1, 2$ is derived for the WG solution. Some numerical experiments are conducted to verify the efficiency and accuracy of our WG method and furthermore a superconvergence has been observed from the numerical results.