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Understanding the Limits of Inf-Sup Stable Galerkin-FEM for Incompressible Flows

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Boundary and Interior Layers, Computational and Asymptotic Methods - BAIL 2014

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 108))

Abstract

The core of numerical simulations of coupled incompressible flow problems consists of a robust, accurate and fast solver for the time-dependent, incompressible Navier-Stokes equations. We consider inf-sup stable finite element methods with grad-div stabilization and symmetric stabilization of local projection type. The approach is based on a proper scale separation and only the small unresolved scales are modeled. Error estimates for the spatially discretized problem with reasonable growth of the Gronwall constant for large Reynolds numbers are given together with a critical discussion of the choice of stabilization parameters. The fast solution of the fully discretized problems (using BDF(2) in time) is accomplished via unconditionally stable velocity-pressure segregation.

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References

  1. Araya, R., Poza, A.H., Valentin, F.: An adaptive residual local projection finite element method for the Navier-Stokes equations. Adv. Comput. Math. 40(5–6), 1093–1119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem. Numer. Methods Part. Diff. Equ. 31(4), 1224–1250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bangerth, W., Hartmann, R., Kanschat, G.: Deal.II – a general-purpose object-oriented finite element library ACM Trans. Math. Softw. 33(4), 24/1–27 (2007)

    Google Scholar 

  4. Bangerth, W., Burstedde, C., Heister, T., Kronbichler, M.: Algorithms and data structures for massively parallel generic adaptive finite element codes. ACM Trans. Math. Softw. 38, 14/1–28 (2011)

    Article  MathSciNet  Google Scholar 

  5. Bangerth, W., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Young, T.D.: The deal.II Library, Version 8.1 (2013). arXiv preprint, http://arxiv.org/abs/1312.2266v4

  6. Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Braack, M., Mucha, P.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burman, E., Fernandez, M.A.: Continuous interior penalty FEM for the time-dependent Navier-Stokes equations. Numer. Math. 107, 39–77 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Couzy, W.: Spectral element discretization of the unsteady Navier-Stokes equations and its iterative solution on parallel computers. Ph.D. Thesis, EPFL Lausanne (1995)

    Google Scholar 

  10. Erturk, E., Corke, T.C., Gokcol, C.: Numerical solutions of 2D-steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48, 747–774 (2005)

    Article  MATH  Google Scholar 

  11. Girault, V., Scott, R.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1–19 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guermond, J.L.: Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl. 88, 87–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guermond, J.L., Minev, P., Shen, J.: Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43(1), 239–258 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babus̃ka-Brezzi condition: a stable Petrov-Galerkin formulation Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986)

    Google Scholar 

  15. Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite elements methods for the incompressible Stokes equations. ESAIM: Math. Model. Numer. Anal. (2015). doi:10.1051/m2an/2015044

    MATH  Google Scholar 

  18. Löwe, J., Lube, G.: A projection-based variational multiscale method for large-eddy simulation with application to non-isothermal free-convection problems. Math. Model. Methods Appl. Sci. 22(2) (2012). doi:10.1142/S0218202511500114

    Google Scholar 

  19. Lube, G., Rapin, G., Löwe, J.: Local projection stabilization for incompressible flows: equal-order vs. inf-sup stable interpolation. Electron. Trans. Numer. Anal. 32, 106–122 (2008)

    Google Scholar 

  20. Matthies, G., Tobiska, L.: Local projection type stabilisation applied to inf-sup stable discretisations of the problem. IMA J. Numer. Anal. (2014). doi:10.1093/imanum/drt064

    MATH  Google Scholar 

  21. Matthies, G., Skrzypacz, P., Tobiska, L.: A unified convergence analysis for local projection stabilisations applied to the Oseen problem. Math. Model. Numer. Anal. 41(4), 713–742 (2007). M2AN 2007

    Google Scholar 

  22. Röhe, L., Lube, G.: Analysis of a variational multiscale method for large-eddy simulation and its application to homogeneous isotropic turbulence. Comput. Methods Appl. Mech. Eng. 199, 2331–2342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin/Heidelberg (2008)

    MATH  Google Scholar 

  24. Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder. Notes Numer. Fluid Mech. 52, 547–566 (1996). In: Hirschel, E.H. (ed.) Flow Simulation with High-performance Computers II, pp. 547–566

    Google Scholar 

  25. Qin, Y., Feng, M., Luo, K., Wu, K.: Local projection stabilized finite element methods for Navier-Stokes equations. Appl. Math. Mech. – Engl. Ed. 31(5), 7651–7664 (2010)

    Google Scholar 

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Acknowledgements

The work of Daniel Arndt was supported by CRC 963 founded by German research council (DFG). The work of Helene Dallmann was supported by the RTG 1023 founded by German research council (DFG).

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Authors and Affiliations

  1. Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestrasse 16-18, D-37083, Göttingen, Germany

    Gert Lube, Daniel Arndt & Helene Dallmann

Authors
  1. Gert Lube
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  2. Daniel Arndt
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  3. Helene Dallmann
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Corresponding author

Correspondence to Gert Lube .

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Editors and Affiliations

  1. Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic

    Petr Knobloch

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Lube, G., Arndt, D., Dallmann, H. (2015). Understanding the Limits of Inf-Sup Stable Galerkin-FEM for Incompressible Flows. In: Knobloch, P. (eds) Boundary and Interior Layers, Computational and Asymptotic Methods - BAIL 2014. Lecture Notes in Computational Science and Engineering, vol 108. Springer, Cham. https://doi.org/10.1007/978-3-319-25727-3_12

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