Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

Yu Zhang; Hai Bi; Yidu Yang

Applications of Mathematics (2021)

  • Volume: 66, Issue: 1, page 1-19
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Zhang, Yu, Bi, Hai, and Yang, Yidu. "Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients." Applications of Mathematics 66.1 (2021): 1-19. <http://eudml.org/doc/297091>.

@article{Zhang2021,
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.},
author = {Zhang, Yu, Bi, Hai, Yang, Yidu},
journal = {Applications of Mathematics},
keywords = {correction; Steklov eigenvalue problem; Crouzeix-Raviart finite element; asymptotic lower bounds; convergence order},
language = {eng},
number = {1},
pages = {1-19},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients},
url = {http://eudml.org/doc/297091},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Zhang, Yu
AU - Bi, Hai
AU - Yang, Yidu
TI - Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 1
EP - 19
AB - 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.
LA - eng
KW - correction; Steklov eigenvalue problem; Crouzeix-Raviart finite element; asymptotic lower bounds; convergence order
UR - http://eudml.org/doc/297091
ER -

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