Nonconforming finite elements for the Brinkman and $-\text{curl}\mathrm{\Delta }\text{curl}$ problems on cubical meshes

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