Volume 29, Issue 4
A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems

Huangxin Chen, Jingzhi Li, Weifeng QiuChao Wang

Commun. Comput. Phys., 29 (2021), pp. 1125-1151.

Published online: 2021-02

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  • Abstract

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains. We impose element-wise stabilization instead of stabilization along mesh interfaces. Thus our scheme can be implemented as easy as standard Nédélec's methods for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.

  • Keywords

Quad-curl problem, mixed finite element scheme, error estimates, eigenvalue problem.

  • AMS Subject Headings

65M60, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1125, author = {Chen , Huangxin and Li , Jingzhi and Qiu , Weifeng and Wang , Chao}, title = {A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {4}, pages = {1125--1151}, abstract = {

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains. We impose element-wise stabilization instead of stabilization along mesh interfaces. Thus our scheme can be implemented as easy as standard Nédélec's methods for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0108}, url = {http://global-sci.org/intro/article_detail/cicp/18649.html} }
TY - JOUR T1 - A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems AU - Chen , Huangxin AU - Li , Jingzhi AU - Qiu , Weifeng AU - Wang , Chao JO - Communications in Computational Physics VL - 4 SP - 1125 EP - 1151 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0108 UR - https://global-sci.org/intro/article_detail/cicp/18649.html KW - Quad-curl problem, mixed finite element scheme, error estimates, eigenvalue problem. AB -

The quad-curl problem arises in the resistive magnetohydrodynamics (MHD) and the electromagnetic interior transmission problem. In this paper we study a new mixed finite element scheme using Nédélec's edge elements to approximate both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains. We impose element-wise stabilization instead of stabilization along mesh interfaces. Thus our scheme can be implemented as easy as standard Nédélec's methods for Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical examples are provided to show the viability and accuracy of the proposed method for quad-curl source problem.

Huangxin Chen, Jingzhi Li, Weifeng Qiu & Chao Wang. (2021). A Mixed Finite Element Scheme for Quad-Curl Source and Eigenvalue Problems. Communications in Computational Physics. 29 (4). 1125-1151. doi:10.4208/cicp.OA-2020-0108
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