  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.
