, 55:3 | Cite as

Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations



We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time \(H^1\) stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.

Mathematics Subject Classification

65M12 65M60 35Q30 


  1. 1.
    Babuska, I.: Error bounds for finite element methods. Numer. Math. 16, 322–333 (1971)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bochev, P.: Negative norm least-squares methods for the velocity–vorticity–pressure Navier–Stokes equations. Numer. Methods Partial Differ. Equ. 15(2), 237–256 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multpliers. R.A.I.R.O 8, 129–151 (1974)MATHGoogle Scholar
  5. 5.
    Charnyi, S., Heister, T., Olshanskii, M., Rebholz, L.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ervin, V.J., Heuer, N.: Approximation of time-dependent, viscoelastic fluid flow: Crank–Nicolson, finite element approximation. Numer. Methods Partial Differ. Equ. 20, 248–283 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)CrossRefMATHGoogle Scholar
  8. 8.
    Gresho, P., Sani, R.: Incompressible Flow and the Finite Element Method, vol. 2. Wiley, New York (1998)MATHGoogle Scholar
  9. 9.
    Gresho, P.M.: On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: theory. Int. J. Numer. Methods Fluids 11(5), 587–620 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Heath, M.: Scientific Computing: An Introductory Survey. McGraw-Hill, New York (2002)MATHGoogle Scholar
  11. 11.
    Heister, T., Olshanskii, M. A., Rebholz, L. G.: Unconditional long-time stability of a velocity–vorticity method for 2D Navier–Stokes equations. Numerische Mathematik 135(1), 143–167 (2016)Google Scholar
  12. 12.
    Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem. Part IV: error analysis for the second order time discretization. SIAM J. Numer. Anal. 2, 353–384 (1990)CrossRefMATHGoogle Scholar
  13. 13.
    John, V.: Large Eddy Simulation of Turbulent Incompressible Flows: Analytical and Numerical Results for a Class of LES Models. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2004)CrossRefGoogle Scholar
  14. 14.
    Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach Science Publishers, New York (1969)MATHGoogle Scholar
  15. 15.
    Layton, W.: Introduction to Finite Element Methods for Incompressible, Viscous Flow. SIAM, Philadelphia (2008)Google Scholar
  16. 16.
    Layton, W., Manica, C.C., Neda, M., Olshanskii, M., Rebholz, L.G.: On the accuracy of the rotation form in simulations of the Navier–Stokes equations. J. Comput. Phys. 228, 3433–3447 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lee, H.K., Olshanskii, M.A., Rebholz, L.G.: On error analysis for the 3D Navier–Stokes equations in velocity–vorticity–helicity form. SIAM J. Numer. Anal. 49(2), 711–732 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liska, R., Wendroff, B.: Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25, 995–1017 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Najjar, F., Vanka, S.: Simulations of the unsteady separated flow past a normal flat plate. Int. J. Numer. Methods Fluids 21, 525–547 (1995)CrossRefMATHGoogle Scholar
  20. 20.
    Olshanskii, M.A., Rebholz, L.G.: Velocity–vorticity–helicity formulation and a solver for the Navier–Stokes equations. J. Comput. Phys. 229, 4291–4303 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Olshanskii, M.A., Heister, T., Rebholz, L., Galvin, K.: Natural vorticity boundary conditions on solid walls. Comput. Methods Appl. Mech. Eng. 297, 18–37 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Saha, A.: Far-wake characteristics of two-dimensional flow past a normal flat plate. Phys. Fluids 19(128110), 1–4 (2007)MATHGoogle Scholar
  23. 23.
    Saha, A.: Direct numerical simulation of two-dimensional flow past a normal flat plate. J. Eng. Mech. 139(12), 1894–1901 (2013)CrossRefGoogle Scholar
  24. 24.
    Schroeder, P., Lube, G.: Pressure-robust analysis of divergence-free and conforming fem for evolutionary incompressible Navier–Stokes flows. J. Numer. Math. (2016).
  25. 25.
    Tezduyar, T., Mittal, S., Ray, S., Shih, R.: Incompressible flow computations with stabilized bilinear and linear equal order interpolation velocity–pressure elements. Comput. Methods Appl. Mech. Eng. 95, 221–242 (1992)CrossRefMATHGoogle Scholar
  26. 26.
    Tone, F.: On the long-time stability of the Crank–Nicholson scheme for the 2D Navier–Stokes equations. Numer. Methods D. E. 23(5), 1235–1248 (2007)CrossRefMATHGoogle Scholar
  27. 27.
    Tone, F., Wirosoetisno, D.: On the long-time stability of the implicit Euler scheme for the two-dimensional Navier–Stokes equations. SIAM J. Numer. Anal. 44(1), 29–40 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wong, K.L., Baker, A.J.: A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm. Int. J. Numer. Methods Fluids 38, 99–123 (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

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