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\(C^0\)IP Methods for the Transmission Eigenvalue Problem

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Abstract

We consider a non-selfadjoint fourth order eigenvalue problem using a discontinuous Galerkin (DG) method. For high order problems, DG methods are competitive since they use simple basis functions and have less degrees of freedom. The numerical implementation is much easier compared with classical conforming finite element methods. In this paper, we propose an interior penalty discontinuous Galerkin method using \(C^0\) Lagrange elements (\(C^0\)IP) for the transmission eigenvalue problem and prove the optimal convergence. The method is applied to several examples and its effectiveness is validated.

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Acknowledgments

X. Ji is supported by the National Natural Science Foundation of China (Nos. 11271018, 91230203) and the Special Funds for National Basic Research Program of China (973 Program 2012CB025904, 863 Program 2012AA01A3094). Research of H. Geng is partly supported by the scholarship from China Scholarship Council (CSC) and the NSFC Grant (11371385) and NSFC Grant (11201506) and the Fundamental Research Funds for the Central Universities Grant (CDJXS12101101). The work of J. Sun is supported in part by NSF DMS-1521555 and the US Army Research Laboratory and the US Army Research Office under the cooperative agreement number W911NF-11-2-0046. The work of L. Xu is partially supported by the NSFC Grant (11371385), the Start-up fund of Youth 1000 plan of China and that of Youth 100 plan of Chongqing University.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Hongrui Geng & Liwei Xu

  2. LSEC, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Xia Ji

  3. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Jiguang Sun

  4. Institute of Computing and Data Sciences, Chongqing University, Chongqing, 400044, China

    Liwei Xu

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  1. Hongrui Geng
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  2. Xia Ji
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Correspondence to Xia Ji.

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Geng, H., Ji, X., Sun, J. et al. \(C^0\)IP Methods for the Transmission Eigenvalue Problem. J Sci Comput 68, 326–338 (2016). https://doi.org/10.1007/s10915-015-0140-2

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  • DOI: https://doi.org/10.1007/s10915-015-0140-2

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