On the Convergence Rate of a Preconditioned Subspace Eigensolver

Abstract.

In this paper we present a proof of convergence for a preconditioned subspace method which shows the dependency of the convergence rate on the preconditioner used. This convergence rate depends only on the condition of the pre-conditioned system $\kappa _{2}(MA)$ and the relative separation of the first two eigenvalues $1-\lambda _{1}/\lambda _{2}$. This means that, for example, multigrid preconditioners can be used to find eigenvalues of elliptic PDE's at a grid-independent rate.