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

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Abstract

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, SIAM J Matrix Anal 31:621–628, 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.

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Authors and Affiliations

  1. Sekretariat MA 3-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Thorsten Rohwedder, Reinhold Schneider & Andreas Zeiser

Authors
  1. Thorsten Rohwedder
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  2. Reinhold Schneider
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  3. Andreas Zeiser
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Correspondence to Reinhold Schneider.

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Communicated by Yuesheng Xu.

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Rohwedder, T., Schneider, R. & Zeiser, A. Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization. Adv Comput Math 34, 43–66 (2011). https://doi.org/10.1007/s10444-009-9141-8

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  • DOI: https://doi.org/10.1007/s10444-009-9141-8

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