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

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 precision 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 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 ingredient 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.
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Pré-publication, Document de travail
2018
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https://hal.inria.fr/hal-01483461
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Soumis le : lundi 16 avril 2018 - 22:06:06
Dernière modification le : jeudi 17 mai 2018 - 09:53:52

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  • HAL Id : hal-01483461, version 2

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Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík. Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework. 2018. 〈hal-01483461v2〉

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