New guaranteed lower bounds on eigenvalues by conforming finite elements
This work provides a practical and implementable approach for obtaining reliable lower eigenvalue bounds in computational mechanics and related fields, addressing a known bottleneck in eigenvalue certification.
The paper presents two new methods for computing guaranteed lower bounds of eigenvalues for symmetric elliptic second-order differential operators with mixed boundary conditions, using a posteriori error estimators based on local flux reconstructions within the standard finite element method. Numerical examples demonstrate practical performance.
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.