Fully <i>H</i> (gradcurl)-nonconforming finite element method for the singularly perturbed quad-curl problem on cubical meshes

Abstract In this paper, we develop a fully nonconforming (both $H(\operatorname{grad}\operatorname{curl})$-nonconforming and $H(\operatorname{curl})$-nonconforming) finite element on cubical meshes that can fit into the Stokes complex. The newly proposed element has 24 degrees of freedom. Unlike the fully $H(\operatorname{grad}\operatorname{curl})$-nonconforming tetrahedral finite elements in Huang (2023, Nonconforming finite element stokes complexes in three dimensions. Sci. China Math., 66, 1879–1902), the element in this paper yields a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order $\mathscr O(h)$ for a fixed parameter $\varepsilon $ and the uniform convergence of order $\mathscr O(h^{1/2})$ for any value of $\varepsilon $ in the sense of an $H(\operatorname{grad}\operatorname{curl})$-norm. When $\varepsilon $ tends to 0, we employ the Nitsche’s method to improve the convergence order from $\mathscr O(h^{1/2})$ to $\mathscr O(h)$. Some numerical examples are used to verify the validity of the theoretical analysis.