Analysis of a projection method for the Stokes problem using an \(\varepsilon \)-Stokes approach

  • Masato Kimura
  • Kazunori MatsuiEmail author
  • Adrian Muntean
  • Hirofumi Notsu
Original Paper


We generalize pressure boundary conditions of an \(\varepsilon \)-Stokes problem. Our \(\varepsilon \)-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter \(\varepsilon >0\). For the Dirichlet boundary condition, it is proven in Matsui and Muntean (Adv Math Sci Appl, 27:181–191, 2018) that the solution for the \(\varepsilon \)-Stokes problem converges to the one for the Stokes problem as \(\varepsilon \) tends to 0, and to the one for the pressure-Poisson problem as \(\varepsilon \) tends to \(\infty \). Here, we extend these results to the Neumann and mixed boundary conditions. We also establish error estimates in suitable norms between the solutions to the \(\varepsilon \)-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in \(\varepsilon \). Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the \(\varepsilon \)-Stokes problem has a nice asymptotic structure.


Stokes problem Pressure-Poisson equation Asymptotic analysis Finite element method 

Mathematics Subject Classification

76D03 35Q35 35B40 65N30 



  1. 1.
    Amsden, A.A., Harlow, F.H.: A simplified MAC technique for incompressible fluid flow calculations. J. Comput. Phys. 6, 322–325 (1970)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chan, R.K.C., Street, R.L.: A computer study of finite-amplitude water waves. J. Comput. Phys. 6, 68–94 (1970)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22(104), 745–762 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conca, C., Murat, F., Pironneau, O.: The Stokes and Navier–Stokes equations with boundary conditions involving the pressure. Jpn. J. Math. 20(2), 279–318 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conca, C., Pares, C., Pironneau, O., Thiriet, M.: Navier–Stokes equations with imposed pressure and velocity fluxes. Int. Numer. Methods. Fluids 20, 267–287 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cummins, S.J., Rudman, M.: An SPH projection method. J. Comput. Phys. 152, 584–607 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Douglas, J., Wang, J.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495–508 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  9. 9.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  10. 10.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 9, pp. 3–1176. North-Holland, Amsterdam (2003)Google Scholar
  11. 11.
    Guermond, J.L., Quartapelle, L.: On stability and convergence of projection methods based on pressure Poisson equation. Int. J. Numer. Methods Fluids 26, 1039–1053 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harlow, F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hughes, T., Franca, L., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59, 85–99 (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, J., Moin, P.: Application of a fractional-step method to incompressible Navier–Stokes equations. J. Comput. Phys. 59, 308–323 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marušić, S.: On the Navier–Stokes system with pressure boundary condition. Ann. Univ. Ferrara 53, 319–331 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Matsui, K., Muntean, A.: Asymptotic analysis of an \(\varepsilon \)-Stokes problem connecting Stokes and pressure-Poisson problems. Adv. Math. Sci. Appl. 27, 181–191 (2018)MathSciNetGoogle Scholar
  18. 18.
    McKee, S., Tomé, M.F., Cuminato, J.A., Castelo, A., Ferreira, V.G.: Recent advances in the marker and cell method. Arch. Comput. Methods Eng. 2, 107–142 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (1967). (Translated in 2012 by Kufner, A. and Tronel, G)zbMATHGoogle Scholar
  20. 20.
    Perot, J.B.: An analysis of the fractional step method. J. Comput. Phys. 108, 51–58 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Temam, R.: Navier–Stokes Equations. North Holland, Amsterdam (1979)zbMATHGoogle Scholar
  22. 22.
    Viecelli, J.A.: A computing method for incompressible flows bounded by moving walls. J. Comput. Phys. 8, 119–143 (1971)CrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsKanazawa UniversityKanazawaJapan
  2. 2.Division of Mathematical and Physical Sciences, Graduate School of Natural Science and TechnologyKanazawa UniversityKanazawaJapan
  3. 3.Department of Mathematics and Computer ScienceKarlstad UniversityKarlstadSweden
  4. 4.Japan Science and Technology Agency, PRESTOKawaguchiJapan

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