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Volume 34, Issue 5
Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes

Qian Zhang, Min Zhang & Zhimin Zhang

DOI: 10.4208/cicp.OA-2023-0102

Commun. Comput. Phys., 34 (2023), pp. 1332-1360.

Published online: 2023-12

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  • Abstract

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.

  • Keywords

Nonconforming elements, −curl∆curl problem, Brinkman problem, finite element de Rham complex, Stokes complex.

  • AMS Subject Headings

65N30, 35Q60, 65N15, 35B45

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-34-1332, author = {Zhang , QianZhang , Min and Zhang , Zhimin}, title = {Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes}, journal = {Communications in Computational Physics}, year = {2023}, volume = {34}, number = {5}, pages = {1332--1360}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0102}, url = {http://global-sci.org/intro/article_detail/cicp/22216.html} }
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 -

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.

Qian Zhang, Min Zhang & Zhimin Zhang. (2023). Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes. Communications in Computational Physics. 34 (5). 1332-1360. doi:10.4208/cicp.OA-2023-0102
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