On the stability of the Rayleigh-Ritz method for eigenvalues

This paper studies global stability properties of the Rayleigh–Ritz approximation of eigenvalues of the Laplace operator. The focus lies on the ratios λ^k/λk of the kth numerical eigenvalue λ^k 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.