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Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations


Authors: Liuqiang Zhong, Long Chen, Shi Shu, Gabriel Wittum and Jinchao Xu
Journal: Math. Comp. 81 (2012), 623-642
MSC (2010): Primary 65F10, 65N30; Secondary 65N12, 78A25
DOI: https://doi.org/10.1090/S0025-5718-2011-02544-5
Published electronically: December 13, 2011
MathSciNet review: 2869030
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a standard Adaptive Edge Finite Element Method (AEFEM) based on arbitrary order Nédélec edge elements, for three-dimen-
sional indefinite time-harmonic Maxwell equations. We prove that the AEFEM gives a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops provided the initial mesh is fine enough. Using the geometric decay, we show that the AEFEM yields the best possible decay rate of the error plus oscillation in terms of the number of degrees of freedom. The main technical contribution of the paper is the establishment of a quasi-orthogonality and a localized a posteriori error estimator.


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Additional Information

Liuqiang Zhong
Affiliation: School of Mathematical and Computational Sciences, Xiangtan University, Hunan 411105, China
Address at time of publication: School of Mathematics Sciences, South China Normal University, Guangzhou 510631, China
Email: zhong@scnu.edu.cn

Long Chen
Affiliation: Department of Mathematics, University of California, Irvine, California 92697
Email: chenlong@math.uci.edu

Shi Shu
Affiliation: School of Mathematical and Computational Sciences, Xiangtan University, Hunan 411105, China
Email: shushi@xtu.edu.cn

Gabriel Wittum
Affiliation: Simulation and Modeling, Goethe Center for Scientific Computing, Goethe-University, Kettenhofweg 139, 60325 Frankfurt am Main, Germany
Email: wittum@gcsc.uni-frankfurt.de

Jinchao Xu
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02544-5
Received by editor(s): June 3, 2010
Published electronically: December 13, 2011
Additional Notes: The first author was supported in part by the National Natural Science Foundation of China (Grant No. 11026091)
The second author was supported by NSF Grant DMS-0811272 and in part by 2010-2011 UC Irvine CORCL
The third author was supported in part by NSFC Key Project 11031006, Hunan Provincial Natural Science Foundation of China (Grant No. 10JJ7001), and the Aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China
The fourth author was supported in part by the German ministry of Economics and technology, BMWi, and the GRS (Gesellschaft für Reaktorsicherheit).
The last author was supported in part by NSF Grant DMS-0915153, DMS-0749202 and the Alexander von Humboldt Research Award for Senior US Scientist.
Article copyright: © Copyright 2011 American Mathematical Society

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