TY - JOUR
T1 - Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes
AU - Zhang , Qian
AU - Zhang , Min
AU - Zhang , Zhimin
JO - Communications in Computational Physics
VL - 5
SP - 1332
EP - 1360
PY - 2023
DA - 2023/12
SN - 34
DO - http://doi.org/10.4208/cicp.OA-2023-0102
UR - https://global-sci.org/intro/article_detail/cicp/22216.html
KW - Nonconforming elements, −curl∆curl problem, Brinkman problem, finite element
de Rham complex, Stokes complex.
AB - <p style="text-align: justify;">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.</p>