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.
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Pré-publication, Document de travail
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Contributeur : Martin Vohralik <>
Soumis le : dimanche 5 mars 2017 - 22:38:44
Dernière modification le : mercredi 21 mars 2018 - 18:57:50
Document(s) archivé(s) le : mardi 6 juin 2017 - 12:22:28


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  • HAL Id : hal-01483461, version 1



Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík. Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework. 2017. 〈hal-01483461〉



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