Sponsor
This work was supported in part by the NSF grant DMS-2245077. It also benefited from activities organized under the auspices of NSF RTG grant DMS-2136228.
Published In
Computational Methods in Applied Mathematics
Document Type
Pre-Print
Publication Date
2-25-2026
Subjects
Error Estimation, Finite Element Methods, Rational Functions
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.
Rights
© Copyright the author(s) 2026
DOI
10.1515/cmam-2025-0161
Persistent Identifier
https://archives.pdx.edu/ds/psu/44537
Citation Details
Published as: Gopalakrishnan, J., & Pinochet-Soto, G. (2026). Reliable Eigenspace Error Estimation Using Source Error Estimators. Computational Methods in Applied Mathematics.
Description
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: Gopalakrishnan, J., & Pinochet-Soto, G. (2026). Reliable eigenspace error estimation using source error estimators. Computational Methods in Applied Mathematics, (0).