Guaranteed a posteriori bounds for eigenvalues and eigenvectors: multiplicities and clusters

TL;DR

This paper develops guaranteed a posteriori error bounds for eigenvalue clusters and eigenvectors of second-order self-adjoint elliptic operators, applicable to degenerate eigenvalues and verified through numerical tests.

Contribution

It introduces a new framework for guaranteed a posteriori bounds on eigenvalues and eigenvectors, including cases with eigenvalue multiplicities and clusters, with practical computability.

Findings

Bounds are valid under eigenvalue separation assumptions.

Bounds converge at the same rate as the true errors.

Numerical tests confirm the effectiveness of the bounds.

Abstract

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schr{\"o}dinger operator with periodic boundary conditions of the form $--$\Delta$ + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.