Eigenvalue enclosures

TL;DR

This paper investigates numerical methods for computing eigenvalue enclosures, analyzing their equivalence, extending theoretical results, and providing convergence estimates supported by benchmark experiments on Maxwell eigenvalues.

Contribution

It establishes the equivalence between two eigenvalue enclosure methods, extends theoretical results, and provides explicit convergence estimates with numerical validation.

Findings

Proved the equivalence of two eigenvalue enclosure methods.

Extended theoretical results with convergence estimates.

Validated methods through Maxwell eigenvalue problem experiments.

Abstract

This paper is concerned with methods for numerical computation of eigenvalue enclosures. We examine in close detail the equivalence between an extension of the Lehmann-Maehly-Goerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We extend various previously known results in the theory and establish explicit convergence estimates in both settings. The theoretical results are supported by two benchmark numerical experiments on the isotropic Maxwell eigenvalue problem.