On the stability of the Rayleigh-Ritz method for eigenvalues

Abstract

This paper studies global stability properties of the Rayleigh-Ritz approximation of eigenvalues of the Laplace operator. The focus lies on the ratios <em>^λ_k</em>/<em>λ_k</em> of the <em>k</em>th numerical eigenvalue <em>^λ_k</em> and the <em>k</em>th exact eigenvalue <em>λ_k</em>. 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.