Lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

TL;DR

This paper develops a new correction method for finite element eigenvalue approximations to establish lower bounds for Steklov eigenvalues with variable coefficients, ensuring convergence from below and validating results numerically.

Contribution

It introduces a novel correction technique for finite element eigenvalue approximations that guarantees lower bounds and preserves convergence order for variable coefficient Steklov problems.

Findings

Corrected eigenvalues provide reliable lower bounds.

Convergence from below is maintained regardless of eigenfunction regularity.

Numerical experiments confirm theoretical predictions.

Abstract

In this paper, using new correction to the Crouzeix-Raviart finite element eigenvalue approximations, we obtain lower eigenvalue bounds for the Steklov eigenvalue problem with variable coefficients on d-dimensional domains (d = 2,3). In addition, we prove that the corrected eigenvalues asymptotically converge to the exact ones from below whether the eigenfunctions are singular or smooth and whether the eigenvalues are large enough or not. Further, we prove that the corrected eigenvalues still maintain the same convergence order as that of uncorrected eigenvalues. Finally, numerical experiments validate our theoretical results.