A Posteriori Error Estimates for Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques

TL;DR

This paper introduces an a posteriori error estimator for high-order finite element methods applied to elliptic eigenvalue problems, effectively estimating errors in eigenvalue clusters and invariant subspaces.

Contribution

It develops a novel error estimator based on auxiliary error functions for high-order finite element discretizations of elliptic eigenvalue problems.

Findings

Estimator accurately measures eigenvalue cluster errors

Effective in estimating subspace gaps

Numerical experiments confirm practical utility

Abstract

We propose an a posteriori error estimator for high-order $p$- or $hp$-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate invariant subspaces. Numerical experiments demonstrate the practical effectivity of the approach.