Applications of Mathematics, first online, pp. 1-19
Asymptotic lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients
Yu Zhang, Hai Bi, Yidu Yang
Received April 26, 2019. Published online September 9, 2020.
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.
Affiliations: Yu Zhang, School of Mathematical Sciences, Guizhou Normal University, No. 116 Baoshan Road (N), Guiyang 550001, China; School of Mathematics & Statistics, Guizhou University of Finance and Economics, Huayan Rd, Huaxi District, Guiyang 550025, China, e-mail: email@example.com; Hai Bi, Yidu Yang (corresponding author), School of Mathematical Sciences, Guizhou Normal University, No. 116 Baoshan Road (N), Guiyang 550001, China, e-mail: firstname.lastname@example.org, email@example.com