This short note shows the superconvergence of an $H(\mathrm{grad}\,\mathrm{curl})$-nonconforming brick element very recently introduced in [17] for the quad-curl problem. The supercloseness is based on proper modifications for both the interpolation and the discrete formulation, leading to an $O(h^2)$ superclose order in the discrete $H(\mathrm{grad}\,\mathrm{curl})$ norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.

翻译：本文简要证明了文献[17]中最新提出的用于四旋度问题的$H(\mathrm{grad}\,\mathrm{curl})$非协调砖形单元的超收敛性。通过适当修改插值格式和离散格式，该超逼近性质使得离散$H(\mathrm{grad}\,\mathrm{curl})$范数达到$O(h^2)$超逼近阶。此外，我们提出了一种合适的后处理方法以确保全局超收敛性。数值结果验证了理论分析。