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Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

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Abstract

A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method.

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Authors and Affiliations

  1. Department of Mathematics, Strathclyde University, 26 Richmond St., Glasgow, G1 1XH, Scotland

    Mark Ainsworth

  2. Institute of Mathematics, Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Tomáš Vejchodský

Authors
  1. Mark Ainsworth
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  2. Tomáš Vejchodský
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Correspondence to Mark Ainsworth.

Additional information

Dedicated to Professor Ivo Babuška on the occasion of his 85th birthday.

Partial support for M.A. from the Engineering and Physical Sciences Research Council under grant GR/S35103 and the support for T.V. from the Czech Science Foundation, from the Grant Agency of the Academy of Sciences, and from the Academy of Sciences of the Czech Republic, projects No. 102/07/0496, No. IAA100760702, and the institutional research plan No. AV0Z10190503 are gratefully acknowledged.

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Ainsworth, M., Vejchodský, T. Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems. Numer. Math. 119, 219–243 (2011). https://doi.org/10.1007/s00211-011-0384-1

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  • DOI: https://doi.org/10.1007/s00211-011-0384-1

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