One feature of eigenvalue problems that complicates the estimation of
error is the possibility of repeated or tightly-clustered eigenvalues,
which arise very naturally in domains with symmetries or
near-symmetries, and will heavily feature in our numerical
experiments. When such eigenvalues are to be approximated in
practice, it makes little sense to try to determine whether computed
eigenvalue approximations that are very close to each other are all
approximating the same (repeated) eigenvalue, or approximating
eigenvalues that just happen to very close to each other. In this
case, it is best to estimate eigenvalue error and associated invariant
subspace error in a ``collective sense''. As is the case with solutions of source
problems (boundary value problems), eigenvectors can have
singularities due to domain geometry and/or discontinuities in the
differential operator or boundary conditions, and the types and
severity of singularities that can occur are
well-understood. Unlike
source problems, where the strongest singular behavior that can be
present is typically seen in practice, with eigenvalue problems, the
regularity of eigenvectors varies (dramatically) depending on where
you are in the spectrum.
Our approach to error estimation in the
eigenvalue context is based on related work for source
problems, in that eigenvector errors are approximated
\textit{as functions} in an auxiliary space that, in a practical
sense, complements the finite element space in which the eigenvectors
are approximated.
Appropriate norms of such approximate error
functions provide the basis for estimating eigenvalue and invariant
subspace errors. Because we compute approximate eigenvector error
functions, we can provide qualitative, as well as quantitative
estimates of error.
The efficacy of our approach is demonstrated with numerical experiments,
where either the spectrum is known analytically or its structure can be constructed
a priori. Of particular interest is the situation where the discretization
of the problem does not necessarily satisfy the standard assumptions for convergence.
We observe the so-called mixing of modes where the eigenpairs appear out-of-order.