On variational eigenvalue approximation of semidefinite operators

TL;DR

This paper develops a geometric framework for analyzing the convergence of variational eigenvalue approximations of semidefinite operators, offering new criteria and practical methods for verification.

Contribution

It introduces the vanishing gap condition as an alternative to discrete compactness for eigenvalue convergence analysis and links it to compatible operators with convergence properties.

Findings

Vanishing gap condition is equivalent to eigenvalue convergence.

Compatible operators with Aubin-Nitsche estimates characterize the vanishing gap.

Examples show certain implications are not equivalences.

Abstract

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an Aubin-Nitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.