Numerische Mathematik

, Volume 137, Issue 2, pp 339–351 | Cite as

On the stability of the Rayleigh–Ritz method for eigenvalues

  • D. Gallistl
  • P. Huber
  • D. PeterseimEmail author


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.

Mathematics Subject Classification

65N12 65N15 65N30 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut für Angewandte und Numerische MathematikKarlsruher Institut für TechnologieKarlsruheGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
  3. 3.Institut für MathematikUniversität AugsburgAugsburgGermany

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