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Numerische Mathematik

, Volume 140, Issue 4, pp 1033–1079 | Cite as

Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework

  • Eric Cancès
  • Geneviève Dusson
  • Yvon Maday
  • Benjamin Stamm
  • Martin VohralíkEmail author
Article

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.

Mathematics Subject Classification

35P15 65N15 65N25 65N30 65N50 

Notes

Acknowledgements

We would like to thank Frédéric Hecht (Laboratoire Jacques-Louis Lions, Sorbonne Université) for his kind help with our higher-order implementation in the FreeFem++ code [42, 43].

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Eric Cancès
    • 1
    • 7
  • Geneviève Dusson
    • 2
  • Yvon Maday
    • 3
    • 4
    • 5
  • Benjamin Stamm
    • 6
  • Martin Vohralík
    • 1
    • 7
    Email author
  1. 1.InriaParisFrance
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions (LJLL)ParisFrance
  4. 4.Institut Universitaire de FranceParisFrance
  5. 5.Division of Applied MathematicsBrown UniversityProvidenceUSA
  6. 6.Center for Computational Engineering ScienceRWTH Aachen UniversityAachenGermany
  7. 7.CERMICS, Ecole des PontsUniversité Paris-EstMarne-la-ValléeFrance

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