  How close are Galerkin eigenvectors to the best approximation available out
of the trial subspace ? Under a variety of conditions the Galerkin method gives
an approximate eigenvector that approaches asymptotically the projection of the
exact eigenvector onto the trial subspace -- and this occurs more rapidly than
the underlying rate of convergence of the approximate eigenvectors. Both
orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a
special emphasis on nonselfadjoint problems. Consequences for the numerical
treatment of elliptic PDEs discretized either with finite element methods or
with spectral methods are discussed and an application to Krylov subspace
methods for large scale matrix eigenvalue problems is presented. New lower
bounds to the $sep$ of a pair of operators are developed as well.

Spectral Theory;Numerical Analysis