Asymptotic Lower Bounds of Eigenvalues for the Steklov Eigenvalue Problem

Abstract

This paper introduces a nonconforming finite element method (also known as a weak Galerkin finite element method) to solve the Steklov eigenvalue problems, focusing on obtaining lower bounds of the eigenvalues. Compared with the existing work, the proposed method can provide asymptotic lower bound approximations of eigenvalues with arbitrary high-order convergences under only the assumptions on the regularities of eigenfunctions. Moreover, another algorithm, which can compute lower bounds of eigenvalues with proper selections of parameters, is presented. Numerical results on the square domain and the L-shaped domain demonstrate the accuracy and lower bound property of the numerical schemes.

Keywords

Finite element method / Steklov eigenvalue problem / Asymptotic lower bound / Guaranteed lower bound / 65N15 / 65N25 / 65N30

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Funding

Key Technologies Research and Development Program(2023YFB3309104)

National Natural Science Foundation of China(1233000214)

Beijing Natural Science Foundation(Z200003)

RIGHTS & PERMISSIONS

Shanghai University