Reliable Eigenspace Error Estimation Using Source Error Estimators

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.