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An Adaptive Finite Element Method for the Transmission Eigenvalue Problem

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Abstract

The classical weak formulation of the Helmholtz transmission eigenvalue problem can be linearized into an equivalent nonsymmetric eigenvalue problem. Based on this nonsymmetric eigenvalue problem, we first discuss the a posteriori error estimates and adaptive algorithm of conforming finite elements for the Helmholtz transmission eigenvalue problem. We give the a posteriori error indicators for primal eigenfunction, dual eigenfunction and eigenvalue. Theoretical analysis shows that the indicators for both primal eigenfunction and dual eigenfunction are reliable and efficient and that the indicator for eigenvalue is reliable. Numerical experiments confirm our theoretical analysis.

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Acknowledgments

We cordially thank the referees and the editor for their valuable comments that led to the large improvement of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

    Jiayu Han & Yidu Yang

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  1. Jiayu Han
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  2. Yidu Yang
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Correspondence to Yidu Yang.

Additional information

Project supported by the National Natural Science Foundation of China (Grant No. 11561014).

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Han, J., Yang, Y. An Adaptive Finite Element Method for the Transmission Eigenvalue Problem. J Sci Comput 69, 1279–1300 (2016). https://doi.org/10.1007/s10915-016-0234-5

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  • DOI: https://doi.org/10.1007/s10915-016-0234-5

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