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Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors : a unified framework
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Cancès, Eric, Dusson, Geneviève, Maday, Yvon, Stamm, Benjamin and Vohralík, Martin (2018) Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors : a unified framework. Numerische Mathematick, 140 (4). pp. 1033-1079. doi:10.1007/s00211-018-0984-0 ISSN 0029-599X.
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Official URL: https://doi.org/10.1007/s00211-018-0984-0
Abstract
This paper develops a general framework for a posteriori error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the energy error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i-th eigenvalue lies between the exact (i−1) -th and (i+1) -th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the accuracy of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredients are equivalences between the i-th eigenvalue error, the associated eigenvector energy error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.
Item Type: | Journal Article | |||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||
Library of Congress Subject Headings (LCSH): | Eigenvalues, Eigenvectors, Finite element method, Galerkin methods | |||||||||||||||
Journal or Publication Title: | Numerische Mathematick | |||||||||||||||
Publisher: | Springer | |||||||||||||||
ISSN: | 0029-599X | |||||||||||||||
Official Date: | 25 December 2018 | |||||||||||||||
Dates: |
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Volume: | 140 | |||||||||||||||
Number: | 4 | |||||||||||||||
Page Range: | pp. 1033-1079 | |||||||||||||||
DOI: | 10.1007/s00211-018-0984-0 | |||||||||||||||
Status: | Peer Reviewed | |||||||||||||||
Publication Status: | Published | |||||||||||||||
Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-0984-0 | |||||||||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
Date of first compliant deposit: | 20 July 2018 | |||||||||||||||
Date of first compliant Open Access: | 25 September 2019 | |||||||||||||||
RIOXX Funder/Project Grant: |
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