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Fully computable a posteriori error bounds for eigenfunctions

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Abstract

For compact self-adjoint operators in Hilbert spaces, two algorithms are proposed to provide fully computable a posteriori error estimate for eigenfunction approximation. Both algorithms apply well to the case of tight clusters and multiple eigenvalues, under the settings of target eigenvalue problems. Algorithm I is based on the Rayleigh quotient and the min-max principle that characterizes the eigenvalue problems. The formula for the error estimate provided by Algorithm I is easy to compute and applies to problems with limited information of Rayleigh quotients. Algorithm II, as an extension of the Davis–Kahan method, takes advantage of the dual formulation of differential operators along with the Prager–Synge technique and provides greatly improved accuracy of the estimate, especially for the finite element approximations of eigenfunctions. Numerical examples of eigenvalue problems of matrices and the Laplace operators over convex and non-convex domains illustrate the efficiency of the proposed algorithms.

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Notes

  1. See https://ganjin.online/xfliu/EigenVecEstimation for source codes and demonstrations of all presented examples.

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Acknowledgements

The authors greatly appreciate the valuable referees’ comments. Thanks to them the original manuscript considerably improved. The first author also show thanks to Dr. Yuji Nakatsukasa from Oxford University for his introduction of Davis–Kahan’s method.

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Authors and Affiliations

  1. Graduate School of Science and Technology, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata City, Niigata, 950-2181, Japan

    Xuefeng Liu

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, Prague 1, 115 67, Czech Republic

    Tomáš Vejchodský

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  1. Xuefeng Liu
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Correspondence to Xuefeng Liu or Tomáš Vejchodský.

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The Xuefeng Liu is supported by Japan Society for the Promotion of Science: Fund for the Promotion of Joint International Research (Fostering Joint International Research (A)) 20KK0306, Grant-in-Aid for Scientific Research (B) 20H01820, 21H00998, and Grant-in-Aid for Scientific Research (C) 18K03411. The Tomas Vejchodsky is supported by the Czech Science Foundation, project no. 20-01074S, and by RVO 67985840. This work is also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center of Kyoto University.

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Liu, X., Vejchodský, T. Fully computable a posteriori error bounds for eigenfunctions. Numer. Math. 152, 183–221 (2022). https://doi.org/10.1007/s00211-022-01304-0

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