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A Posteriori Error Estimates for Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques

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Abstract

We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate invariant subspaces. Numerical experiments demonstrate the practical effectivity of the approach.

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Authors and Affiliations

  1. Department of Engineering, Durham University, South Road, Durham, DH1 3LE, United Kingdom

    Stefano Giani

  2. Department of Mathematics, University of Zagreb, Bijenička 30, 10000, Zagreb, Croatia

    Luka Grubišić

  3. Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland

    Harri Hakula

  4. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, 464R Fariborz Maseeh Hall, Portland, OR, 97201, USA

    Jeffrey S. Ovall

Authors
  1. Stefano Giani
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  2. Luka Grubišić
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  3. Harri Hakula
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  4. Jeffrey S. Ovall
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Correspondence to Jeffrey S. Ovall.

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Grubišić: The work of this author was supported by Hrvatska Zaklada za Znanost (Croatian Science Foundation) under the grant IP-2019-04-6268 - Randomized low rank algorithms and applications to parameter dependent problems.

Hakula: The work of this author was supported by the (FP7/2007–2013) ERC grant agreement no 339380

Ovall: The work of this author was supported by the National Science Foundation under contract DMS-1522471.

All: The authors gratefully acknowledge the Mathematisches Forschungsinstitut Oberwolfach for hosting them through the Research-In-Pairs program on the topic “High-Order Finite Element Methods for Elliptic Eigenvalue Problems”

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Giani, S., Grubišić, L., Hakula, H. et al. A Posteriori Error Estimates for Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques. J Sci Comput 88, 55 (2021). https://doi.org/10.1007/s10915-021-01572-2

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  • DOI: https://doi.org/10.1007/s10915-021-01572-2

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