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Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements

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Abstract

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier–Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier–Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order \(\mathcal O(h^{2})\) in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. Both the continuous-in-time case and the fully discrete scheme with the backward Euler method as time integrator are analyzed.

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Authors and Affiliations

  1. Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Valladolid, Spain

    Javier de Frutos

  2. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Bosco García-Archilla

  3. Weierstrass Institute for Applied Analysis and Stochastics, Leibniz Institute in Forschungsverbund Berlin e. V. (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    Volker John

  4. Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195, Berlin, Germany

    Volker John

  5. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Julia Novo

Authors
  1. Javier de Frutos
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  2. Bosco García-Archilla
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  3. Volker John
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  4. Julia Novo
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Corresponding author

Correspondence to Julia Novo.

Additional information

Communicated by: Karsten Urban

Javier de Frutos research was supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE).

Bosco García-Archilla research was supported by Spanish MINECO under grant MTM2015-65608-P. Julia Novo research was supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE).

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de Frutos, J., García-Archilla, B., John, V. et al. Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements. Adv Comput Math 44, 195–225 (2018). https://doi.org/10.1007/s10444-017-9540-1

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  • DOI: https://doi.org/10.1007/s10444-017-9540-1

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