We propose two families of nonconforming elements on cubical meshes: one for the $-\text{curl}\Delta\text{curl}$ problem and the other for the Brinkman problem. The element for the $-\text{curl}\Delta\text{curl}$ problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the parameter $\nu$. The lowest-order elements for the $-\text{curl}\Delta\text{curl}$ and the Brinkman problems have 48 and 30 degrees of freedom, respectively. The two families of elements are subspaces of $H(\text{curl};\Omega)$ and $H(\text{div};\Omega)$, and they, as nonconforming approximation to $H(\text{gradcurl};\Omega)$ and $[H^1(\Omega)]^3$, can form a discrete Stokes complex together with the Lagrange element and the $L^2$ element.

翻译：本文提出了两类立方体网格上的非协调元：一类用于$-\text{curl}\Delta\text{curl}$问题，另一类用于Brinkman问题。用于$-\text{curl}\Delta\text{curl}$问题的单元是立方体网格上的首个非协调元。用于Brinkman问题的单元能够针对参数$\nu$生成一致稳定的有限元方法。$-\text{curl}\Delta\text{curl}$问题和Brinkman问题的最低阶单元分别具有48个和30个自由度。这两类单元族分别是$H(\text{curl};\Omega)$和$H(\text{div};\Omega)$的子空间，并且作为$H(\text{gradcurl};\Omega)$和$[H^1(\Omega)]^3$的非协调逼近，它们可与Lagrange单元和$L^2$单元共同构成离散Stokes复形。