Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization

摘要

In this paper we discuss an abstract iteration scheme for the calculation ofthe smallest eigenvalue of an elliptic operator eigenvalue problem. A short andgeometric proof based on the preconditioned inverse iteration (PINVIT) formatrices [Knyazev and Neymeyr, (2009)] is extended to the case of operators. Weshow that convergence is retained up to any tolerance if one only usesapproximate applications of operators which leads to the perturbedpreconditioned inverse iteration (PPINVIT). We then analyze the Besovregularity of the eigenfunctions of the Poisson eigenvalue problem on apolygonal domain, showing the advantage of an adaptive solver to uniformrefinement 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 thepredicted behaviour.