Three methods for two-sided bounds of eigenvalues - a comparison

TL;DR

This paper compares three finite element methods for obtaining two-sided bounds of eigenvalues of symmetric elliptic operators, evaluating their accuracy, efficiency, and applicability through numerical examples.

Contribution

It provides a comprehensive comparison of Lehmann-Goerisch, Crouzeix-Raviart, and combined Weinstein-Kato methods for eigenvalue bounds.

Findings

Lehmann-Goerisch method shows high accuracy in bounds.

Crouzeix-Raviart method offers computational efficiency.

Combined bounds are versatile for various eigenvalue orders.

Abstract

We compare three finite element based methods designed for two-sided bounds of eigenvalues of symmetric elliptic second order operators. The first method is known as the Lehmann-Goerisch method. The second method is based on Crouzeix-Raviart nonconforming finite element method. The third one is a combination of generalized Weinstein and Kato bounds with complementarity based estimators. We concisely describe these methods and use them to solve three numerical examples. We compare their accuracy, computational performance, and generality in both the lowest and higher order case.