Reliable eigenspace error estimation using source error estimators

TL;DR

The paper develops a framework to bound the gap between exact and discretized eigenspaces for clusters of eigenvalues of unbounded nonselfadjoint operators with compact resolvent by repurposing source-problem error estimators through rational spectral mappings. A key theoretical contribution is the error-representation approach that reduces eigenspace error to a small, computable source-estimator problem, yielding a cluster-robust bound $ ext{gap}_V(E_h,E) \,\le\, C\| \mathcal{E}\widehat{e}_h\|_Y$ based on a finite-dimensional generalized eigenproblem. The framework is instantiated for FE discretizations using two residual-based schemes, FOSLS and DPG, with proofs of resolvent-approximation convergence and reliability of the estimators. Numerical experiments on a Gordon-Webb-Wolpert drum and a Bragg fiber demonstrate cluster-focused adaptive refinement and show that the DPG-based estimators provide effective, tighter control of eigenspace error compared to standard residual-based estimators. Overall, the work enables practical, adaptive estimation of eigenvalue-cluster gaps in challenging nonselfadjoint settings, with direct applicability to FEAST-type spectral filters and related methods.

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.

Figures (9)

• Figure 1: Eigenfunctions corresponding to the eighth, ninth and tenth eigenvalues ($\lambda_8$, $\lambda_9$ and $\lambda_{10}$) of the Laplace operator a GWW isospectral drum. Scale varies between figures.
• Figure 2: On the left, initial mesh on a GWW isospectral drum. On the center and right, final meshes after the adaptive algorithm, using the DPG error estimator for computing the single eigenvalue $\lambda_9$ (center) and the cluster $\Lambda_\mathsf{cl}$ (right).
• Figure 3: Hausdorff distance between the (approximated) ninth eigenvalue $\Lambda_\mathsf{sg} = \{\lambda_9\}$ and its computed approximation, $d_{\bullet}$, and $\ell^2$-norm of the error estimator $\eta_{\ell^2,\bullet}$, for the different discretizations and error estimators ($\bullet \in \{\mathsf{cg-res}, \mathsf{cg-dwr}, \mathsf{dpg}\}$), against the number of dofs. The dashed red line indicates the level of accuracy of the reference eigenvalues.
• Figure 4: Efficiency ratio for the single eigenvalue approximation $\Lambda_\mathsf{sg} = \{\lambda_9\}$, with different discretizations and error estimators ($\bullet \in \{\mathsf{cg-res}, \mathsf{cg-dwr}, \mathsf{dpg}\}$), against the number of dofs. (dofs beyond 12,000 have been omitted for clarity.)
• Figure 5: Hausdorff distance between the (approximated) cluster of known eigenvalues $\Lambda_\mathsf{cl} = \{\lambda_8, \lambda_9, \lambda_{10}\}$ and the computed eigenvalues, $d_{\bullet}$, and $\ell^2$-norm of the error estimator $\eta_{\ell^2,\bullet}$, for the different discretizations and error estimators ($\bullet \in \{\mathsf{cg-res}, \mathsf{cg-dwr}, \mathsf{dpg}\}$), against the number of dofs. The dashed red line indicates the level of accuracy of the reference eigenvalues.

Theorems & Definitions (23)

• Lemma 2.1
• proof
• Example 2.2: A finite-dimensional case
• Example 2.3: The Cayley transform of an operator
• Remark 2.4: General rational functions
• Lemma 3.2
• proof
• Example 3.3
• Example 3.4: The Cayley transform revisited
• Example 3.5: Rational function in inverse iterations