On reduced basis methods for eigenvalue problems, and on its coupling with perturbation theory

TL;DR

This paper analyzes reduced basis methods for eigenvalue problems, providing error bounds and coupling with perturbation theory, especially for parameter-dependent operators, supported by numerical examples.

Contribution

It introduces new error bounds for eigenvalue approximations and extends analysis to degenerate and parameter-dependent cases.

Findings

Error bounds between exact and approximate eigenmodes

Analysis of degenerate eigenvalue cases

Numerical validation of theoretical results

Abstract

In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact eigenmodes and the approximated ones and also consider degenerate cases in the analysis. When the operator depends on a parameter, we apply the bounds assuming that the reduced space contains the derivatives of the eigenfunction with respect to the parameter. Finally, we provide some numerical examples that reflect the analytical results.