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Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm

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Abstract

We consider the approximation in the reaction-diffusion norm with continuous finite elements and prove that the best error is equivalent to a sum of the local best errors on pairs of elements. The equivalence constants do not depend on the ratio of diffusion to reaction. We illustrate the usefulness of this result with two applications. First, we discuss robustness and locking properties of continuous finite elements with respect to the reaction-diffusion norm. Second, we derive local error functionals that ensure robust performance of adaptive tree approximation in the reaction-diffusion norm.

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Acknowledgments

We thank Peter Binev for a fruitful discussion on Theorem 8.1 as well as the anonymous referees for their critical remarks that helped us to improve a previous version of this work. Part of F. Tantardini’s work was founded by the DFG under grant AOBJ:612415.

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Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44801, Bochum, Germany

    Francesca Tantardini & Rüdiger Verfürth

  2. Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Andreas Veeser

Authors
  1. Francesca Tantardini
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  2. Andreas Veeser
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  3. Rüdiger Verfürth
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Correspondence to Andreas Veeser.

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Communicated by Wolfgang Dahmen.

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Tantardini, F., Veeser, A. & Verfürth, R. Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm. Constr Approx 42, 313–347 (2015). https://doi.org/10.1007/s00365-015-9291-5

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  • DOI: https://doi.org/10.1007/s00365-015-9291-5

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