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Mathematics of Computation

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

Authors: Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm and Martin Vohralík
Journal: Math. Comp. 89 (2020), 2563-2611
MSC (2010): Primary 35P15, 65N15, 65N25, 65N30
Published electronically: July 30, 2020
MathSciNet review: 4136540
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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 $-\Delta + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

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Additional Information

Eric Cancès
Affiliation: CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France; and Inria, 2 Rue Simone Iff, 75589 Paris, France

Geneviève Dusson
Affiliation: Université Bourgogne Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France
ORCID: 0000-0002-7160-6064

Yvon Maday
Affiliation: Sorbonne Université and Université de Paris, CNRS, Laboratoire Jacques-Louis Lions (LJLL), 75005 Paris, France; and Institut Universitaire de France, 75005 Paris, France
MR Author ID: 117765

Benjamin Stamm
Affiliation: Center for Computational Engineering Science, RWTH Aachen University, Aachen, Germany
MR Author ID: 824171
ORCID: 0000-0003-3375-483X

Martin Vohralík
Affiliation: Inria, 2 Rue Simone Iff, 75589 Paris, France; and CERMICS, Ecole des Ponts ParisTech, 6 & 8 Av. Pascal, 77455 Marne-la-Vallée, France
ORCID: 0000-0002-8838-7689

Received by editor(s): May 13, 2019
Received by editor(s) in revised form: March 4, 2020
Published electronically: July 30, 2020
Additional Notes: Part of this work has been supported from French state funds managed by the CalSimLab LABEX and the ANR within the “Investissements d’Avenir” program (reference ANR-11-LABX-0037-01). The last author has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 647134 GATIPOR). Part of this work was supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-0003).
The first and third authors acknowledge funding from PICS- CNRS, PHC PROCOPE 2017 (Project No. 37855ZK), and European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367).
Article copyright: © Copyright 2020 American Mathematical Society