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Robust error estimates for approximations of non-self-adjoint eigenvalue problems

Numerische Mathematik volume 133, pages 471–495 (2016)Cite this article

Abstract

We present new residual estimates based on Kato’s square root theorem for spectral approximations of non-self-adjoint differential operators of convection–diffusion–reaction type. It is not assumed that the eigenvalue/vector approximations are obtained from any particular numerical method, so these estimates may be applied quite broadly. Key eigenvalue and eigenvector error results are illustrated in the context of an hp-adaptive finite element algorithm for spectral computations, where it is shown that the resulting a posteriori error estimates are reliable. The efficiency of these error estimates is also strongly suggested empirically.

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Acknowledgments

The authors would like to thank Prof. Dr. V. Mehrmann, Technical University Berlin, for very helpful comments on the manuscript and Dr. C. Engström, University of Umea for bringing the early and important references [30, 31] to our attention. We also thank the referees and editor for very helpful suggestions on refining the manuscript. L. Grubišić was supported by the Croatian MZOS Grant Nr. 037-0372783-2750 “Spectral decompositions—numerical methods and applications” and the bilateral MZOS–NSF Grant “Estimates for finite element approximation error by auxiliary subspace method”. A. Międlar was supported by the DFG Research Center Matheon. J. Ovall was supported by the National Science Foundation under contract DMS-1414365.

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Authors and Affiliations

  1. School of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LE, UK

    Stefano Giani

  2. Department of Mathematics, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Luka Grubišić

  3. Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, Germany

    Agnieszka Międlar

  4. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 315 Neuberger Hall, Portland, OR, 97201, USA

    Jeffrey S. Ovall

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  1. Stefano Giani
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  2. Luka Grubišić
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  3. Agnieszka Międlar
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  4. Jeffrey S. Ovall
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Correspondence to Luka Grubišić.

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Giani, S., Grubišić, L., Międlar, A. et al. Robust error estimates for approximations of non-self-adjoint eigenvalue problems. Numer. Math. 133, 471–495 (2016). https://doi.org/10.1007/s00211-015-0752-3

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  • DOI: https://doi.org/10.1007/s00211-015-0752-3

Mathematics Subject Classification

  • 65N30
  • 65N25
  • 65N15