Geodesic
Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients
Applications of Mathematics, Tome 66 (2021) no. 1, pp. 1-19.

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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.
DOI : 10.21136/AM.2020.0108-19
Classification : 65N25, 65N30
Keywords: correction; Steklov eigenvalue problem; Crouzeix-Raviart finite element; asymptotic lower bounds; convergence order 
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Zhang, Yu; Bi, Hai; Yang, Yidu. Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients. Applications of Mathematics, Tome 66 (2021) no. 1, pp. 1-19. doi : 10.21136/AM.2020.0108-19. http://geodesic.mathdoc.fr/articles/10.21136/AM.2020.0108-19/

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