P000020

*Harri Hakula (Aalto University)

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.

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Keywords | eigenproblems, clusters, hp-fem |