On variational eigenvalue approximation of semidefinite operators

• Main Authors: Christiansen, Snorre Harald, Winther, Ragnar
• Format: Journal Article
• Language: English
• Published: 26-10-2011
• Subjects: Mathematics - Analysis of PDEs; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.1005.2059

IMA J. Numer. Anal., Vol. 33, No. 1, p. 164-189, 2013 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.