Jump to ContentJump to Main Navigation
Show Summary Details
More options ÔÇŽ

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, Fr├ęd├ęric / 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

IMPACT FACTOR 2018: 3.107

CiteScore 2018: 2.43

SCImago Journal Rank (SJR) 2018: 1.252
Source Normalized Impact per Paper (SNIP) 2018: 1.618

Mathematical Citation Quotient (MCQ) 2018: 1.13

See all formats and pricing
More options ÔÇŽ
Volume 25, Issue 4


On three steps two-grid finite element methods for the 2D-transient Navier-Stokes equations

Saumya Bajpai / Amiya K. Pani
  • Corresponding author
  • Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-02-22 | DOI: https://doi.org/10.1515/jnma-2016-1055


In this paper, an error analysis of a three steps two level Galerkin finite element method for the two dimensional transient NavierÔÇôStokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh ­Łĺ»H with mesh size H. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, uH, which is similar to NewtonÔÇÖs type iteration and the resulting linear system is solved on a finer mesh ­Łĺ»h with mesh size h. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in LÔł×(L2)-norm, when h = ­Łĺ¬ (H2Ôłĺ╬┤) and in LÔł×(H1)-norm, when h = ­Łĺ¬(H4Ôłĺ╬┤) for the velocity and in LÔł×(L2)-norm, when h = ­Łĺ¬(H4Ôłĺ╬┤) for the pressure are established for arbitrarily small ╬┤ > 0. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then, based on backward Euler method, a completely discrete scheme is analyzed and a priori error estimates are derived. Results obtained in this paper are sharper than those derived earlier by two-grid methods. Finally, the paper is concluded with some numerical experiments.

Keywords: two-grid method; 2D Navier-Stokes system; semidiscrete scheme; backward Euler method; optimal error estimates; order of convergence; uniform-in-time estimates; uniqueness assumption; numerical experiments

MSC 2010: 65M60; 65M12; 65M15; 35D05; 35D10


  • [1]

    H. Abboud and T. Sayah, A full discretization of the time-dependent NavierÔÇôStokes equations by a two-grid scheme, M2AN Math. Model. Numer. Anal. 42 (2008), 141ÔÇô174.CrossrefGoogle┬áScholar

  • [2]

    H. Abboud, V. Girault, and T. Sayah, A second order accuracy in time for a full discretized time-dependent NavierÔÇôStokes equations by a two-grid scheme, Numer. Math. 114 (2009), 189ÔÇô231.Web┬áof┬áScienceCrossrefGoogle┬áScholar

  • [3]

    A. A. O. Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the NavierÔÇôStokes equations, Numer. Math. 68 (1994), 189ÔÇô213.CrossrefGoogle┬áScholar

  • [4]

    A. Blanca, J. de Frutos, and J. Novo, Improving the accuracy of the mini-element approximation to NavierÔÇôStokes equations,IMA J. Numer. Anal. 27 (2007), 198ÔÇô218.Web┬áof┬áScienceCrossrefGoogle┬áScholar

  • [5]

    A. Blanca, B. Garc├şa-Archilla, and J. Novo, The postprocessed mixed finite-element method for the NavierÔÇôStokes equations,SIAM J. Numer. Anal. 43 (2005), 1091ÔÇô1111.CrossrefWeb┬áof┬áScienceGoogle┬áScholar

  • [6]

    X. Dai and X. Cheng, A two-grid method based on Newton iteration for the NavierÔÇôStokes equations,J. Comput. Appl. Math. 220 (2008), 566ÔÇô573.Web┬áof┬áScienceCrossrefGoogle┬áScholar

  • [7]

    F. A. Fairag, A Two-level finite element discretization for the stream function form of the NavierÔÇôStokes equations, Comput. Math. Appl. 36 (1998), 117ÔÇô127.CrossrefGoogle┬áScholar

  • [8]

    R. C. Feng and M. Y. Chen, Two-grid error estimates for the stream function form of NavierÔÇôStokes equations, Appl. Math. Mech. 23 (2002), 773ÔÇô782.CrossrefGoogle┬áScholar

  • [9]

    J. de Frutos, B. Garc├şa-Archilla, and J. Novo, Optimal error bounds for two-grid schemes applied to the NavierÔÇôStokes equations, Appl. Math. Comput. 218(2012),7034ÔÇô7051.Web┬áof┬áScienceGoogle┬áScholar

  • [10]

    J. de Frutos, B. Garc├şa-Archilla, and J. Novo, Static two-grid mixed finite-element approximations to the NavierÔÇôStokes equations, J. Sci. Comput. 52(2012),619ÔÇô637.CrossrefWeb┬áof┬áScienceGoogle┬áScholar

  • [11]

    B. Garcia-Arch├şlla and E. S. Titi, Postprocessing the Galerkin method: the finite element case, SIAM J. Numer. Anal. 37(2000), 470ÔÇô499.Google┬áScholar

  • [12]

    B. Garcia-Arch├şlla, J. Novo, and E. S. Titi, An approximate inertial manifolds approach to postprocessing the Galerkin method for the NavierÔÇôStokes equations,Math. Comp. 68(1999),893ÔÇô911.CrossrefGoogle┬áScholar

  • [13]

    V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady NavierÔÇôStokes problem in polyhedra, Portugal. Math. 58(2001),25ÔÇô57.Google┬áScholar

  • [14]

    V. Girault and J.-L. Lions, Two-grid finite-element schemes for the transient NavierÔÇôStokes problem,M2AN Math. Model. Numer. Anal. 35 (2001), 945ÔÇô980.CrossrefGoogle┬áScholar

  • [15]

    Y. He, Two-level method based on finite element and CrankÔÇôNicolson extrapolation for the time-dependent NavierÔÇôStokes equations,SIAM J. Numer. Anal. 41(2003),1263ÔÇô1285.CrossrefGoogle┬áScholar

  • [16]

    J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary NavierÔÇôStokes problem: I. Regularity of solutions and second order error estimates for spatial discretization, SIAM J. Numer. Anal. 19(1982), 275ÔÇô311.CrossrefGoogle┬áScholar

  • [17]

    J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary NavierÔÇôStokes problem: IV. Error analysis for second-order time discretization,SIAM J. Numer. Anal. 27(1990),353ÔÇô384.CrossrefGoogle┬áScholar

  • [18]

    A. T. Hill and E. S├╝li, Approximation of the global attractor for the incompressible NavierÔÇôStokes equations, IMA J. Numer. Anal. 20(2000),633ÔÇô667.CrossrefGoogle┬áScholar

  • [19]

    L. G. Margolin, E. S. Titi, and S. Wynne, The postprocessing Galerkin and nonlinear Galerkin methods ÔÇô a truncation analysis point of view,SIAM J. Numer. Anal. 41(2003),695ÔÇô714.CrossrefGoogle┬áScholar

  • [20]

    W. Layton and L. Tobiska, A two-level method with backtracking for the NavierÔÇôStokes equations, SIAM J. Numer. Anal. 35(1998),2035ÔÇô2054.CrossrefGoogle┬áScholar

  • [21]

    W. Layton and W. Lenferink, Two-level Picard and modified Picard methods for the NavierÔÇôStokes equations, Appl. Math. Comput. 69(1995),263ÔÇô274.Google┬áScholar

  • [22]

    W. Layton and H. W. J. Lenferink, A multilevel mesh independence principle for the NavierÔÇôStokes equations, SIAM J. Numer. Anal. 33(1996),17ÔÇô30.CrossrefGoogle┬áScholar

  • [23]

    W. Layton, A two-level discretization method for the NavierÔÇôStokes equations, Comput. Math. Appl. 26(1993),33ÔÇô38.CrossrefGoogle┬áScholar

  • [24]

    Q. Liu and Y. Hou, A two-level finite element method for the NavierÔÇôStokes equations based on a new projection, Appl. Math. Model. 34(2010),383ÔÇô399.CrossrefWeb┬áof┬áScienceGoogle┬áScholar

  • [25]

    M.A. Olshanskii, Two-level method and some a priori estimates in unsteady NavierÔÇôStokes calculations, J. Comput. Appl. Math. 104(1999),173ÔÇô191.CrossrefGoogle┬áScholar

  • [26]

    R. Rannacher, Stable finite element solutions to nonlinear parabolic problems of NavierÔÇôStokes type, Computing methods in applied sciences and engineering, V (Versailles, 1981), North-Holland, Amsterdam, 1982, pp. 301ÔÇô309.Google┬áScholar

  • [27]

    R. Temam, NavierÔÇôStokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, 1984.Google┬áScholar

About the article

Received: 2016-06-10

Revised: 2017-01-29

Accepted: 2017-01-29

Published Online: 2018-02-22

Published in Print: 2017-12-20

Citation Information: Journal of Numerical Mathematics, Volume 25, Issue 4, Pages 199ÔÇô228, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI:┬áhttps://doi.org/10.1515/jnma-2016-1055.

Export Citation

┬ę 2017 Walter de Gruyter Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the ÔÇťCitation AlertÔÇŁ on the top of this page.

Neela Nataraj and A. S. Vasudeva Murthy
Indian Journal of Pure and Applied Mathematics, 2019, Volume 50, Number 3, Page 739
Lei Wang, Jian Li, and Pengzhan Huang
International Communications in Heat and Mass Transfer, 2018, Volume 98, Page 183

Comments (0)

Please log in or register to comment.
Log in