Superconvergence of a Nonconforming Brick Element for the Quad-Curl Problem

Abstract

This paper shows the superconvergence of an [Formula: see text](grad curl)-nonconforming brick element, which is very recently introduced in Wang et al. [Fully [Formula: see text](grad curl)-nonconforming finite element method for the singularly perturbed quad-curl problem on cubical meshes, preprint (2023), arXiv:2301.03172] for the quad-curl problem. The supercloseness is based on proper modifications for both the interpolation and the discrete formulation, leading to superclose [Formula: see text] in the discrete grad, curl norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.