Simultaneous Reduced Basis Approximation of Parameterized Elliptic Eigenvalue Problems

TL;DR

This paper develops a reduced basis framework for efficiently approximating multiple eigenvalues in parameterized elliptic eigenvalue problems, with focus on error estimation, greedy algorithms, and handling parameter-dependent multiple eigenvalues, relevant for vibro-acoustics.

Contribution

It introduces a novel reduced basis approach for simultaneous eigenvalue approximation, including new error estimators and greedy strategies for parameter-dependent multiple eigenvalues.

Findings

Effective greedy algorithms for eigenvalue approximation

Reliable a posteriori error estimators for eigenvalues

Application potential in vibro-acoustics

Abstract

The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics.