  In this paper we discuss an abstract iteration scheme for the calculation of
the smallest eigenvalue of an elliptic operator eigenvalue problem. A short and
geometric proof based on the preconditioned inverse iteration (PINVIT) for
matrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. We
show that convergence is retained up to any tolerance if one only uses
approximate applications of operators which leads to the perturbed
preconditioned inverse iteration (PPINVIT). We then analyze the Besov
regularity of the eigenfunctions of the Poisson eigenvalue problem on a
polygonal domain, showing the advantage of an adaptive solver to uniform
refinement when using a stable wavelet base. A numerical example for PPINVIT,
applied to the model problem on the L-shaped domain, is shown to reproduce the
predicted behaviour.

Numerical Analysis