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A natural nonconforming FEM for the Bingham flow problem is quasi-optimal

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Abstract

This paper introduces a novel three-field formulation for the Bingham flow problem and its two-dimensional version named after Mosolov together with low-order discretizations: a nonconforming for the classical formulation and a mixed finite element method for the three-field model. The two discretizations are equivalent and quasi-optimal in the sense that the \(H^1\) error of the primal variable is bounded by the error of the \(L^2\) best-approximation of the stress variable. This improves the predicted convergence rate by a log factor of the maximal mesh-size in comparison to the first-order conforming finite element method in a model scenario. Despite that numerical experiments lead to comparable results, the nonconforming scheme is proven to be quasi-optimal while this is not guaranteed for the conforming one.

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Acknowledgments

The third named author was supported by the Berlin Mathematical School.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    C. Carstensen

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, Korea

    C. Carstensen

  3. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    B. D. Reddy

  4. Institut für Numerische Simulation, Universität Bonn, Wegelerstraße 6, 53115, Bonn, Germany

    M. Schedensack

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  1. C. Carstensen
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  2. B. D. Reddy
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  3. M. Schedensack
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Correspondence to C. Carstensen.

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Carstensen, C., Reddy, B.D. & Schedensack, M. A natural nonconforming FEM for the Bingham flow problem is quasi-optimal. Numer. Math. 133, 37–66 (2016). https://doi.org/10.1007/s00211-015-0738-1

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  • DOI: https://doi.org/10.1007/s00211-015-0738-1

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