D. Gallistl, P. Huber, and D. Peterseim.
On the stability of the Rayleigh-Ritz method for eigenvalues.
Numerische Mathematik, 137(2):339-351, Oct 2017.
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This paper studies global stability properties of the Rayleigh-Ritz approximation of eigenvalues of the Laplace operator. The focus lies on the ratios \lambdak/λk of the kth numerical eigenvalue \lambdak and the kth exact eigenvalue λk. 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.
Keywords: isogeometric analysis, eigenvalues, inverse inequality