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

TL;DR

This paper introduces two novel nonconforming finite element families on cubical meshes for complex PDEs, including the first nonconforming element for the $- ext{curl}\Delta ext{curl}$ problem, enhancing stability and compatibility in numerical simulations.

Contribution

The paper presents the first nonconforming element for the $- ext{curl}\Delta ext{curl}$ problem and develops a stable finite element method for the Brinkman problem on cubical meshes.

Findings

First nonconforming element for $- ext{curl}\Delta ext{curl}$ problem.

Stable finite element method for Brinkman problem.

Elements form a discrete Stokes complex.

Abstract

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.