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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

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Volume 25, Issue 4


A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier–Stokes equations

Alexander Linke / Michael Neilan / Leo G. Rebholz / Nicholas E. Wilson
Published Online: 2018-02-19 | DOI: https://doi.org/10.1515/jnma-2016-1024


We prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier–Stokes equations equipped with grad–div stabilization with parameter γ converge to the associated coupled method solution with rate γ−1 as γ → ∞. We prove this result for both backward Euler schemes and BDF2 schemes. Furthermore, we simplify classical numerical analysis of projection methods, allowing us to remove some unnecessary assumptions, such as convexity of the domain. Several numerical experiments are given which verify the convergence rate, and show that projection methods with large grad–div stabilization parameters can dramatically improve accuracy.

Keywords: Navier–Stokes; penalty-projection method; convergence; divergence-free finite elements

MSC 2010: 65M60; 76D05


  • [1]

    Ph. Angot, M. Jobelin and J. Latche, Error analysis of the penalty-projection method for the time dependent Stokes equations, Int. J. Finite Vol. 6 (2009), 1–26.Google Scholar

  • [2]

    D. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, In: Advances in Computer Methods for Partial Differential Equations VII (Eds. R. Vichnevetsky, D. Knight and G. Richter), IMACS, pp. 28–34, 1992.Google Scholar

  • [3]

    M. Benzi, G. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica 14 (2005), 1–137.CrossrefGoogle Scholar

  • [4]

    M. Benzi and M. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comp. 28 (2006), 2095–2113.CrossrefGoogle Scholar

  • [5]

    M. Braack, Outflow conditions for Navier–Stokes equations with skew-symmetric formulation of the convective term, In: Boundary and interior layers, computational and asymptotic methods (Ed. P. Knobloch), BAIL, pp. 35–46, 2015.Google Scholar

  • [6]

    D. Brown, R. Cortez and M. Minion, Accurate projection methods for the incompressible Navier–Stokes equations, J. Comp. Phys. 168 (2001), 464–499.CrossrefGoogle Scholar

  • [7]

    M. Case, V. Ervin, A. Linke and L. Rebholz, A connection between Scott–Vogelius elements and grad–div stabilization, SIAM J. Numer. Anal. 49 (2011), 1461–1481.CrossrefGoogle Scholar

  • [8]

    A. J. Chorin, Numerical solution for the Navier–Stokes equations, Math. Comp. 22 (1968), 745–762.CrossrefGoogle Scholar

  • [9]

    H. Elman, D. Silvester and A. Wathen, Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2005.Google Scholar

  • [10]

    J. Guermond, Un résultat de convergence ďordre deux en temps pour ľapproximation des équations de Navier–Stokes par une technique de projection incrémentale, Math. Modelling Numer. Anal. 33 (1999), 169–189.CrossrefGoogle Scholar

  • [11]

    J. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg. 195 (2006), 6011–6045.CrossrefGoogle Scholar

  • [12]

    J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangulations, Math. Comp. 83 (2014), 15–36.Google Scholar

  • [13]

    J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. Part IV: Error analysis for the second order time discretization, SIAM J. Numer. Anal. 2 (1990), 353–384.Google Scholar

  • [14]

    E. Jenkins, V. John, A. Linke and L. Rebholz, On the parameter choice in grad–div stabilization for the Stokes equations, Adv. Comp. Math. 40 (2014), 491–516.CrossrefGoogle Scholar

  • [15]

    M. Jobelin, C. Lapuerta, J.-C. Latche, Ph. Angot and B. Piar, A finite element penalty-projection method for incompressible flows, J. Comp. Phys. 217 (2006), 502–518.CrossrefGoogle Scholar

  • [16]

    W. Layton, An Introduction to the Numerical Analysis of Viscous Incompressible Flows, SIAM, Philadelphia, 2008.Google Scholar

  • [17]

    W. Layton, C. Manica, M. Neda, M. A. Olshanskii and L. Rebholz, On the accuracy of the rotation form in simulations of the Navier–Stokes equations, J. Comp. Phys. 228 (2009), 3433–3447.CrossrefGoogle Scholar

  • [18]

    W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational testing of a high accuracy Leraydeconvolution model of turbulence, Numer. Meth. Partial Differ. Equ. 24 (2008), 555–582.CrossrefGoogle Scholar

  • [19]

    H. K. Lee, M. A. Olshanskii and L. G. Rebholz, On error analysis for the 3D Navier–Stokes equations in velocity–vorticity– helicity form, SIAM J. Numer. Anal. 49 (2011), 711–732.CrossrefWeb of ScienceGoogle Scholar

  • [20]

    M. A. Olshanskii and A. Reusken, Grad–Div stabilization for the Stokes equations, Math. Comp. 73 (2004), 1699–1718.Google Scholar

  • [21]

    M. A. Olshanskii and E. E. Tyrtyshnikov, Iterative Methods for Linear Systems: Theory and Applications, SIAM, Philadelphia, 2014.Google Scholar

  • [22]

    J. Perot, An analysis of the fractional step method, J. Comp. Phys. 108 (1993), 51–58.CrossrefGoogle Scholar

  • [23]

    A. Prohl, Projection and quasi-compressibility methods for solving the incompressible Navier–Stokes equations, Teubner-Verlag, Stuttgart, 1997.Google Scholar

  • [24]

    A. Prohl, On pressure approximation via projection methods for nonstationary incompressible Navier–Stokes equations, SIAM J. Numer. Anal. 47 (2008), 158–180.Web of ScienceGoogle Scholar

  • [25]

    J. Schöberl, Robust Multigrid Methods for a Parameter Dependent Problem in primal variables, Numer. Alg. 84 (1999), 97–119.Google Scholar

  • [26]

    L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, Math. Modelling Numer. Anal. 19 (1985), 111–143.CrossrefGoogle Scholar

  • [27]

    J. Shen, On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations, Numer. Math. 62 (1992), 49–73.CrossrefGoogle Scholar

  • [28]

    R. Temam, Sur ľapproximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires, II, Arch. Rational Mech. Anal. 33 (1969), 377–385.Google Scholar

  • [29]

    S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp. 74 (2005), 543–554.Google Scholar

About the article

Supported by the DFG Research Center Matheon, project D27

Partially supported by NSF grant DMS1417980 and the Alfred Sloan Foundation

Partially supported by NSF grant DMS1522191

Received: 2016-04-02

Revised: 2016-06-15

Accepted: 2016-07-03

Published Online: 2018-02-19

Published in Print: 2017-12-20

Citation Information: Journal of Numerical Mathematics, Volume 25, Issue 4, Pages 229–248, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnma-2016-1024.

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