Superconvergence of a nonconforming brick element for the quad-curl problem

This short note shows the superconvergence of an $H(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$-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 $H(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$ norm. Moreover, we propose a suitable postprocessing method to ensure the global superconvergence. Numerical results verify our theory.