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Asymptotic Lower Bounds for Eigenvalues of the Steklov Eigenvalue Problem with Variable Coefficients

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Abstract

In this paper, using a new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain asymptotic lower bounds of eigenvalues for the Steklov eigenvalue problem with variable coefficients on d-dimensional domains (d = 2, 3). In addition, we prove that the corrected eigenvalues converge to the exact ones from below. The new result removes the conditions of eigenfunction being singular and eigenvalue being large enough, which are usually required in the existing arguments about asymptotic lower bounds. Further, we prove that the corrected eigenvalues still maintain the same convergence order as uncorrected eigenvalues. Finally, numerical experiments validate our theoretical results.

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematical Sciences, Guizhou Normal University, No. 116 Baoshan Road (N), Guiyang, 550001, China

    Yu Zhang, Hai Bi & Yidu Yang

  2. School of Mathematics & Statistics, Guizhou University of Finance and Economics, Huayan Rd, Huaxi District, Guiyang, 550025, China

    Yu Zhang

Authors
  1. Yu Zhang
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  2. Hai Bi
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  3. Yidu Yang
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Corresponding author

Correspondence to Yidu Yang.

Additional information

This work is supported by the Young Scientific and Technical Talents Development of Education Department of Guizhou Province (KY [2018]153), the National Natural Science Foundation of China (Grant No. 11561014 and No. 11761022).

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Zhang, Y., Bi, H. & Yang, Y. Asymptotic Lower Bounds for Eigenvalues of the Steklov Eigenvalue Problem with Variable Coefficients. Appl Math 66, 1–19 (2021). https://doi.org/10.21136/AM.2020.0108-19

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  • DOI: https://doi.org/10.21136/AM.2020.0108-19

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