A new family of nonconforming elements with $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$-continuity for the three-dimensional quad-curl problem

We propose and analyze a new family of nonconforming finite elements for the three-dimensional quad-curl problem. The proposed finite element spaces are subspaces of $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$, but not of $H(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$, which are different from the existing nonconforming ones. The well-posedness of the discrete problem is proved and optimal error estimates in discrete $H(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$ norm, $H(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l})$ norm and $L^2$ norm are derived. Numerical experiments are provided to illustrate the good performance of the method and confirm our theoretical predictions.