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A Posteriori Estimates Using Auxiliary Subspace Techniques

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Abstract

A posteriori error estimators based on auxiliary subspace techniques for second order elliptic problems in \(\mathbb {R}^d\ (d\ge 2)\) are considered. In this approach, the solution of a global problem is utilized as the error estimator. As the continuity and coercivity of the problem trivially leads to an efficiency bound, the main focus of this paper is to derive an analogous effectivity bound and to determine the computational complexity of the auxiliary approximation problem. With a carefully chosen auxiliary subspace, we prove that the error is bounded above by the error estimate up to oscillation terms. In addition, we show that the stiffness matrix of the auxiliary problem is spectrally equivalent to its diagonal. Several numerical experiments are presented verifying the theoretical results.

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Acknowledgements

This work was supported in part by the National Science Foundation through Grants DMS-1417980 (Neilan) and DMS-1414365 (Ovall).

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Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Aalto University, Aalto, Finland

    Harri Hakula

  2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA

    Michael Neilan

  3. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, USA

    Jeffrey S. Ovall

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  1. Harri Hakula
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  2. Michael Neilan
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  3. Jeffrey S. Ovall
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Correspondence to Jeffrey S. Ovall.

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Hakula, H., Neilan, M. & Ovall, J.S. A Posteriori Estimates Using Auxiliary Subspace Techniques. J Sci Comput 72, 97–127 (2017). https://doi.org/10.1007/s10915-016-0352-0

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  • DOI: https://doi.org/10.1007/s10915-016-0352-0

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