Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

TL;DR

This paper introduces a method combining conforming and nonconforming finite element techniques to efficiently compute tight lower and upper bounds for Laplace eigenvalues, validated by numerical experiments.

Contribution

It presents a novel approach that simultaneously computes eigenvalue bounds using a single eigenvalue problem and specific finite element methods.

Findings

Successfully computes lower bounds using enriched nonconforming elements.

Achieves upper bounds through conforming finite element postprocessing.

Numerical results confirm the theoretical accuracy of the bounds.

Abstract

This article is devoted to computing the lower and upper bounds of the Laplace eigenvalue problem. By using the special nonconforming finite elements, i.e., enriched Crouzeix-Raviart element and extension $Q_1^{\rm rot}$, we get the lower bound of the eigenvalue. Additionally, we also use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue. The postprocessing method need only to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented. Thus, we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once. Some numerical results are also presented to validate our theoretical analysis.