Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes (Communications in Computational Physics, 2023)
We propose two families of nonconforming elements on cubical meshes: onefor the −curl∆curl problem and the other for the Brinkman problem. The elementfor the −curl∆curl problem is the first nonconforming element on cubical meshes.The element for the Brinkman problem can yield a uniformly stable finite elementmethod 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 $H({\rm div};Ω),$ andthey, as nonconforming approximation to $H({\rm gradcurl};Ω)$ and $[H^1(Ω)]^3,$ can form adiscrete Stokes complex together with the serendipity finite element space and thepiecewise polynomial space.