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

Mathematics of Computation, vol. 89, pp. 2563–2611

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 Schrödinger operator with periodic boundary conditions of the form − Δ + V -\Delta + V discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

Authors 5

1. Éric Cancès

   Institut national de recherche en sciences et technologies du numérique · École nationale des ponts et chaussées · CERMICS

   Affiliation as printed

   CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France

   Inria, 2 Rue Simone Iff, 75589 Paris, France

2. Geneviève Dusson

   Centre National de la Recherche Scientifique · Université Bourgogne Franche-Comté · Laboratoire de Mathématiques de Besançon

   Affiliation as printed

   Université Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France

3. Yvon Maday

   Centre National de la Recherche Scientifique · Institut Universitaire de France · Université Paris Cité · Sorbonne Université · Laboratoire Jacques-Louis Lions

   Affiliation as printed

   Sorbonne Université and Université de Paris, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France; and Institut Universitaire de France, 75005 Paris, France

   Institut Universitaire de France, 75005 Paris, France

   Sorbonne Université and Université de Paris, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France

4. Benjamin Stamm Aachen Center for Computational Engineering Science

   RWTH Aachen University

   Affiliation as printed

   Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany

5. Martin Vohralı́k

   Institut national de recherche en sciences et technologies du numérique · École nationale des ponts et chaussées · CERMICS

   Affiliation as printed

   CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France

   Inria, 2 Rue Simone Iff, 75589 Paris, France