Volume 1, Issue 4
A Family of Curl-Curl Conforming Finite Elements on Tetrahedral Meshes

Qian Zhang & Zhimin Zhang

CSIAM Trans. Appl. Math., 1 (2020), pp. 639-663.

Published online: 2020-12

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  • Abstract

In [23], we, together with our collaborator, proposed a family of $H$(curl$^2$)- conforming elements on both triangular and rectangular meshes. The elements provide a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct $H$(curl$^2$)-conforming finite elements on tetrahedral meshes. The newly proposed elements have been proved to have the optimal interpolation error estimate. Having the tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by the conforming finite element method. We also provide several numerical examples of using our elements to solve the quad-curl problem. The results of the numerical experiments show the correctness of our elements.

  • Keywords

$H^2$(curl)-conforming, finite elements, tetrahedral mesh, quad-curl problems, interpolation errors, convergence analysis.

  • AMS Subject Headings

65N30, 35Q60, 65N15, 41A25

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-1-639, author = {Qian and Zhang and and 10032 and and Qian Zhang and Zhimin and Zhang and and 10033 and and Zhimin Zhang}, title = {A Family of Curl-Curl Conforming Finite Elements on Tetrahedral Meshes}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {4}, pages = {639--663}, abstract = {

In [23], we, together with our collaborator, proposed a family of $H$(curl$^2$)- conforming elements on both triangular and rectangular meshes. The elements provide a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct $H$(curl$^2$)-conforming finite elements on tetrahedral meshes. The newly proposed elements have been proved to have the optimal interpolation error estimate. Having the tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by the conforming finite element method. We also provide several numerical examples of using our elements to solve the quad-curl problem. The results of the numerical experiments show the correctness of our elements.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0023}, url = {http://global-sci.org/intro/article_detail/csiam-am/18540.html} }
TY - JOUR T1 - A Family of Curl-Curl Conforming Finite Elements on Tetrahedral Meshes AU - Zhang , Qian AU - Zhang , Zhimin JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 639 EP - 663 PY - 2020 DA - 2020/12 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0023 UR - https://global-sci.org/intro/article_detail/csiam-am/18540.html KW - $H^2$(curl)-conforming, finite elements, tetrahedral mesh, quad-curl problems, interpolation errors, convergence analysis. AB -

In [23], we, together with our collaborator, proposed a family of $H$(curl$^2$)- conforming elements on both triangular and rectangular meshes. The elements provide a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct $H$(curl$^2$)-conforming finite elements on tetrahedral meshes. The newly proposed elements have been proved to have the optimal interpolation error estimate. Having the tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by the conforming finite element method. We also provide several numerical examples of using our elements to solve the quad-curl problem. The results of the numerical experiments show the correctness of our elements.

Qian Zhang & Zhimin Zhang. (2020). A Family of Curl-Curl Conforming Finite Elements on Tetrahedral Meshes. CSIAM Transactions on Applied Mathematics. 1 (4). 639-663. doi:10.4208/csiam-am.2020-0023
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