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A multilevel Newton’s method for the Steklov eigenvalue problem

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Abstract

This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on the coarsest mesh and a sequence of augmented linear problems on refined meshes, derived by Newton step. We prove that this iteration scheme obtains the optimal convergence rate with linear complexity, which improves the overall efficiency of solving the Steklov eigenvalue problem. Moreover, an adaptive iteration scheme for multi eigenvalues based on this new multilevel method is given. Finally, some numerical experiments are provided to illustrate the efficiency of the proposed multilevel scheme.

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Acknowledgements

The authors would like to thank Prof. Hehu Xie (Chinese Academy of Sciences) for his stimulating discussions and fruitful cooperation that have motivated this work. They also would like to thank the anonymous referees for their careful reading and valuable comments and suggestions that helped to improve the contents and presentation of this paper.

Funding

The work of the second author (F. Xu) was supported by the National Natural Science Foundation of China (11801021). The work of the third author (M. Xie) was supported by the National Natural Science Foundation of China (12001402, 12071343).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Technology and Business University, Beijing, 100048, China

    Meiling Yue

  2. Beijing Institute for Scientific and Engineering Computing, College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Fei Xu

  3. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Manting Xie

Authors
  1. Meiling Yue
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  2. Fei Xu
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  3. Manting Xie
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Correspondence to Manting Xie.

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The authors declare no competing interests.

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Communicated by: Long Chen

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Yue, M., Xu, F. & Xie, M. A multilevel Newton’s method for the Steklov eigenvalue problem. Adv Comput Math 48, 33 (2022). https://doi.org/10.1007/s10444-022-09934-6

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  • DOI: https://doi.org/10.1007/s10444-022-09934-6

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