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Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes

Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes

Year:    2023

Author:    Qian Zhang, Min Zhang, Zhimin Zhang

Communications in Computational Physics, Vol. 34 (2023), Iss. 5 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2023-0102

Communications in Computational Physics, Vol. 34 (2023), Iss. 5 : pp. 1332–1360

Published online:    2023-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    29

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

Author Details

Qian Zhang Email

Min Zhang Email

Zhimin Zhang Email

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