On the stability of the Rayleigh-Ritz method for eigenvalues. (English) Zbl 1378.65175
This article deals with the ratio \(\hat{\lambda}_k/\lambda_k\), where \(\hat{\lambda}_k\) (resp. \(\lambda_k\)) is the \(k\)th numerical eigenvalue using the Rayleigh-Ritz methods (resp. the \(k\)th exact eigenvalue of Laplace operator). It is shown that in the context of classical finite elements, the maximal ratio blows up with the polynomial degree. In the context of B-splines of maximum smoothness, it is proved that 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.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P15 | Estimates of eigenvalues in context of PDEs |
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