Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes

We propose two families of nonconforming elements on cubical meshes: one for the −curl∆curl problem and the other for the Brinkman problem. The element for the −curl∆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 viscosity coefficient $ν.$ The lowest-order elements for the −curl∆curl and the Brinkman problems have 48 and 30 DOFs on each cube, respectively. The two families of elements are subspaces of $H({\rm curl};Ω)$ and $H({\rm div};Ω),$ and they, as nonconforming approximation to $H({\rm gradcurl};Ω)$ and $[H^1 (Ω)]^3,$ can form a discrete Stokes complex together with the serendipity finite element space and the piecewise polynomial space.