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How Many Numerical Eigenvalues Can We Trust?

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Abstract

When using finite element and finite difference methods to approximate eigenvalues of 2mth-order elliptic problems, the number of reliable numerical eigenvalues can be estimated in terms of the total degrees of freedom \(N\) in resulting discrete systems. The truth is worse than what we used to believe in that the percentage of reliable eigenvalues decreases with an increased \(N\), even though the number of reliable eigenvalues increases with \(N\).

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Acknowledgments

The author would like to thank Professor Huiyuan Li for producing the two graphs in the paper.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100084, China

    Zhimin Zhang

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Zhimin Zhang
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Correspondence to Zhimin Zhang.

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This work is supported in part by the National Natural Science Foundation of China under Grants 11471031 and 91430216, and the US National Science Foundation through Grants DMS-1115530 and DMS-1419040.

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Zhang, Z. How Many Numerical Eigenvalues Can We Trust?. J Sci Comput 65, 455–466 (2015). https://doi.org/10.1007/s10915-014-9971-5

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  • DOI: https://doi.org/10.1007/s10915-014-9971-5

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