Reliable eigenspace error estimation using source error estimators

TL;DR

This paper develops a reliable error estimator framework for eigenspaces of nonselfadjoint operators, enabling effective adaptive refinement for eigenvalue clusters in complex problems.

Contribution

It introduces a novel framework to estimate eigenspace errors from source problem estimators, applicable to nonselfadjoint operators and their discretizations.

Findings

The estimator bounds eigenspace gaps reliably and is computable.

Applications to FOSLS and DPG discretizations yield new error gap estimators.

Numerical experiments demonstrate effective adaptive refinement targeting eigenvalue clusters.

Abstract

We introduce a framework for repurposing error estimators for source problems to compute an estimator for the gap between eigenspaces and their discretizations. Of interest are eigenspaces of finite clusters of eigenvalues of unbounded nonselfadjoint linear operators with compact resolvent. Eigenspaces and eigenvalues of rational functions of such operators are studied as a first step. Under an assumption of convergence of resolvent approximations in the operator norm and an assumption on global reliability of source problem error estimators, we show that the gap in eigenspace approximations can be bounded by a globally reliable and computable error estimator. Also included are applications of the theoretical framework to first-order system least squares (FOSLS) discretizations and discontinuous Petrov-Galerkin (DPG) discretizations, both yielding new estimators for the error gap. Numerical experiments with a selfadjoint model problem and with a leaky nonselfadjoint waveguide eigenproblem show that adaptive algorithms using the new estimators give refinement patterns that target the cluster as a whole instead of individual eigenfunctions.