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On the stability of the Rayleigh–Ritz method for eigenvalues

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Abstract

This paper studies global stability properties of the Rayleigh–Ritz approximation of eigenvalues of the Laplace operator. The focus lies on the ratios \(\hat{\lambda }_k/\lambda _k\) of the kth numerical eigenvalue \(\hat{\lambda }_k\) and the kth exact eigenvalue \(\lambda _k\). In the context of classical finite elements, the maximal ratio blows up with the polynomial degree. For B-splines of maximum smoothness, the ratios are uniformly bounded with respect to the degree except for a few instable numerical eigenvalues which are related to the presence of essential boundary conditions. These phenomena are linked to the inverse inequalities in the respective approximation spaces.

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Authors and Affiliations

  1. Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, 76128, Karlsruhe, Germany

    D. Gallistl

  2. Institut für Numerische Simulation, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    P. Huber

  3. Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159, Augsburg, Germany

    D. Peterseim

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  1. D. Gallistl
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  2. P. Huber
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  3. D. Peterseim
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Correspondence to D. Peterseim.

Additional information

D. Gallistl is supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. D. Peterseim and P. Huber are supported by DFG in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis” under the project “Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials” (PE2143/2-1). The authors acknowledge the support given by the Hausdorff Center for Mathematics Bonn.

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Gallistl, D., Huber, P. & Peterseim, D. On the stability of the Rayleigh–Ritz method for eigenvalues. Numer. Math. 137, 339–351 (2017). https://doi.org/10.1007/s00211-017-0876-8

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  • DOI: https://doi.org/10.1007/s00211-017-0876-8

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