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An Adaptive FEM for a Maxwell Interface Problem

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Abstract

This paper develops an adaptive edge finite element method for a Maxwell interface problem: \( {\mathbf{curl}}\mu ^{-1} {\mathbf{curl}}u_\delta +\delta \varepsilon u_\delta =f, \) where \(\delta >0\) is allowed to degenerate to zero. A residual-based a posteriori error estimator is analyzed, with \(\delta \)-uniform lower and (global and local) upper error bounds. \(\delta \)-uniform convergence and optimality of the adaptive algorithm are also established. Numerical results are also presented.

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Notes

  1. \(\hat{z}\) is obtained by multiplying the transposed Jacobian matrix of the affine mapping to the composition of z and the affine mapping (see (5.33) in [39, p. 129]).

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Collaborative Innovation Centre of Mathematics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

    Huoyuan Duan

  2. Bank of China, Beijing, 100190, China

    Fengjuan Qiu

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Singapore

    Roger C. E. Tan

  4. LSEC, NCMIS, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Weiying Zheng

Authors
  1. Huoyuan Duan
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  2. Fengjuan Qiu
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  3. Roger C. E. Tan
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  4. Weiying Zheng
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Correspondence to Weiying Zheng.

Additional information

Huoyuan Duan was partially supported by the National Natural Science Foundation of China under the Grants 11071132, 11171168, 91430106, and 11571266, and the Research Fund for the Doctoral Program of Higher Education of China under Grants 20100031110002 and 20120031110026, and the Scientific Research Foundation for Returned Scholars, Ministry of Education of China, and the Wuhan University start-up fund (2042014kf0218) from the Fundamental Research Funds for the Central Universities. Weiying Zheng was supported in part by National Natural Science Foundation of China under the grants11171334 and 91430215, by the Funds for Creative Research Groups of China 11321061, by National 863 Project of China under the Grant 2012AA01A309, and by the National Magnetic Confinement Fusion Science Program 2015GB110003.

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Duan, H., Qiu, F., Tan, R.C.E. et al. An Adaptive FEM for a Maxwell Interface Problem. J Sci Comput 67, 669–704 (2016). https://doi.org/10.1007/s10915-015-0098-0

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  • DOI: https://doi.org/10.1007/s10915-015-0098-0

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