(untitled) (Communications in Computational Physics, 2023)
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(curl;Ω) and H(div;Ω), and they, as nonconforming approximation to H(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.