Guaranteed Eigenvalue Bounds for the Steklov Eigenvalue Problem

Abstract

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed. Compared with the existing algorithm, which deals with eigenvalue problems formulated by positive definite bilinear forms, the newly proposed method can be used to solve the problem defined with positive semidefinite bilinear forms. In practical eigenvalue estimation for the Steklov eigenvalue problem, the Crouzeix--Raviart finite element method along with quantitative error estimation is adopted. Numerical experiments for eigenvalue problems defined on a square domain and an L-shaped domain are provided to validate the precision of computed eigenvalue bounds.