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A nonconforming finite element method for fourth order curl equations in $ \mathbb{R}^3$


Authors: Bin Zheng, Qiya Hu and Jinchao Xu
Journal: Math. Comp. 80 (2011), 1871-1886
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-2011-02480-4
Published electronically: March 25, 2011
MathSciNet review: 2813342
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Abstract: In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem involving both $ (\nabla\times)^2$ and $ (\nabla\times)^4$ operators. The element has a very small number of degrees of freedom, and it imposes the inter-element continuity along the tangential direction which is appropriate for the approximation of magnetic fields. We also provide explicit formulae of basis functions for this element.


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Additional Information

Bin Zheng
Affiliation: Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Address at time of publication: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912
Email: bin_zheng@brown.edu

Qiya Hu
Affiliation: LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
Email: hqy@lsec.cc.ac.cn

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02480-4
Received by editor(s): January 29, 2010
Received by editor(s) in revised form: July 20, 2010
Published electronically: March 25, 2011
Additional Notes: The second author was supported by The Key Project of Natural Science Foundation of China G11031006, National Basic Research Program of China No. G2011309702 and Natural Science Foundation of China G10771178.
The third author was supported by the National Science Foundation under contracts DMS-0609727 and DMS-0915153 and by the Center for Computational Mathematics and Applications, Pennsylvania State University.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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