A reduced order model for the finite element approximation of eigenvalue problems

TL;DR

This paper introduces a reduced order method using POD and a fictitious time parameter to efficiently approximate eigenvalues of Laplace problems, with theoretical guidance on basis size and confirmed accuracy for the first eigenvalue.

Contribution

It presents a novel reduced order approach combining POD and a fictitious time technique, with theoretical basis selection and validated computational results.

Findings

Optimal dimension of POD basis determined theoretically

Approximate solutions accurately capture the first eigenvalue

Method confirms efficiency and accuracy for eigenvalue approximation

Abstract

In this paper we consider a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition. We use a time continuation technique that consists in the introduction of a fictitious time parameter. We use a POD approach and we present some theoretical results showing how to choose the optimal dimension of the POD basis. The results of our computations, related to the first eigenvalue, confirm the optimal behavior of our approximate solution.