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Optimal and Pressure-Independent \(L^2\) Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions

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Book cover Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 77))

Abstract

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a-priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete \(H^1\) velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure-independent \(L^2\) velocity error. Numerical examples confirm the analytical results.

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References

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  3. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

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  4. Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. 268, 782–800 (2014)

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Acknowledgments

This research has been partially funded by the project “Macroscopic Modeling of Transport and Reaction Processes in Magnesium-Air-Batteries” (Grant 03EK3027D) under the research initiative “Energy storage” of the German Federal government.

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Authors and Affiliations

  1. Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistr. 101, 8092, Zürich, Switzerland

    Christian Brennecke

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    Alexander Linke & Christian Merdon

  3. TU Wien, Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8-10/101, 1040, Wien, Austria

    Joachim Schöberl

Authors
  1. Christian Brennecke
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  2. Alexander Linke
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  3. Christian Merdon
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  4. Joachim Schöberl
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Corresponding author

Correspondence to Alexander Linke .

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Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Fuhrmann

  2. Institute Comp. Applied Mathematics, University of Münster Center for Nonlinear Sciences (CeNoS ), Münster, Germany

    Mario Ohlberger

  3. Inst. Appl. Analysis and Num. Simulation, University of Stuttgart, Stuttgart, Germany

    Christian Rohde

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Brennecke, C., Linke, A., Merdon, C., Schöberl, J. (2014). Optimal and Pressure-Independent \(L^2\) Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects. Springer Proceedings in Mathematics & Statistics, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-05684-5_14

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