New guaranteed lower bounds on eigenvalues by conforming finite elements

TL;DR

This paper introduces two new, easily implementable methods for computing guaranteed lower bounds on eigenvalues of symmetric elliptic operators, enhancing accuracy and efficiency in finite element analysis.

Contribution

It generalizes Weinstein's and Kato's bounds to a finite element context, providing practical a posteriori error estimators for eigenvalue lower bounds.

Findings

The methods produce reliable lower bounds for eigenvalues.

They are suitable for adaptive mesh refinement.

Numerical examples demonstrate their effectiveness.

Abstract

We provide two new methods for computing lower bounds of eigenvalues of symmetric elliptic second-order differential operators with mixed boundary conditions of Dirichlet, Neumann, and Robin type. The methods generalize ideas of Weinstein's and Kato's bounds and they are designed for a simple and straightforward implementation in the context of the standard finite element method. These lower bounds are obtained by a posteriori error estimators based on local flux reconstructions, which can be naturally utilized for adaptive mesh refinement. We derive these bounds, prove that they estimate the exact eigenvalues from below, and illustrate their practical performance by a numerical example.