Skip to main content
SpringerLink
Log in

Analysis of time discretization methods for Stokes equations with a nonsmooth forcing term

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider a time dependent Stokes problem that is motivated by two-phase incompressible flow problems with surface tension. The surface tension force results in a right-hand side functional in the momentum equation with poor regularity properties. As a strongly simplified model problem we treat a Stokes problem with a similar time dependent nonsmooth forcing term. We consider the implicit Euler and Crank-Nicolson methods for time discretization. The regularity properties of the data are such that for the Crank-Nicolson method one can not apply error analyses known in the literature. We present a convergence analysis leading to a second order error bound in a suitable negative norm that is weaker that the \(L^2\)-norm. Results of numerical experiments are shown that confirm the analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Chessa, J., Belytschko, T.: An extended finite element method for two-phase fluids. ASME J. Appl. Mech. 70, 10–17 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    Book  MATH  Google Scholar 

  3. Fries, T., Belytschko, T.: The generalized/extended finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84, 253–304 (2010)

    MATH  MathSciNet  Google Scholar 

  4. Girault, V., Raviart, P.: Finite element methods for Navier–Stokes equations. In: Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

  5. Girault, V., Raviart, P.-A.: Finite element approximation of the Navier–Stokes equations. In: Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1979,1981)

  6. Gross, S., Reusken, A.: An extended pressure finite element space for two-phase incompressible flows with surface tension. J. Comp. Phys. 224, 40–58 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gross, S., Reusken, A.: Numerical methods for two-phase incompressible flows. In: Springer series in Computational Mathematics, vol. 40. Springer, Berlin (2011)

  8. Hansbo, A.: Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space. Calcolo 36, 75–101 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem, IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lubich, C., Ostermann, A.: Interior estimates for time discretization of parabolic equations. Appl. Numer. Math. 18, 241–251 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Luskin, M., Rannacher, R.: On the smoothing property of the Crank-Nicolson scheme. Appl. Anal. 14, 117–135 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pillapakkam, S., Singh, P.: A level-set method for computing solutions to viscoelastic two-phase flow. J. Comp. Phys. 174, 552–578 (2001)

    Article  MATH  Google Scholar 

  13. Rannacher, R.: Finite element solution of diffusion problems with irregular. Numer. Math. 43, 309–327 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rannacher, R.: On the numerical solution of the incompressible Navier–Stokes equations. Z. Angew. Math. Mech. 73, 203–216 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Reusken, A.: Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Vis. Sci. 11, 293–305 (2008)

    Article  MathSciNet  Google Scholar 

  16. Solonnikov, V.: \(L_q\)-estimates for a solution to the problem about the evolution of an isolated amount of fluid. J. Math. Sci. (N.Y.) 117, 4237–4259 (2003)

    Article  MathSciNet  Google Scholar 

  17. Solonnikov, V.: On the linear problem arising in the study of a free boundary problem for the Navier–Stokes equations. St. Petersburg Math. J 22, 1023–1049 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sussman, M., Almgren, A., Bell, J., Colella, P., Howell, L., Welcome, M.: An adaptive level set approach for incompressible two-phase flows. J. Comp. Phys. 148, 81–124 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comp. Phys. 114, 146–159 (1994)

    Article  MATH  Google Scholar 

  20. Temam, R.: Navier–Stokes equations. In: Studies in Mathematics and its Applications, vol. 2. North-Holland publishing company, Amsterdam (1984)

  21. Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  22. Turek, S.: Efficient solvers for incompressible flow problems: an algorithmic and computational approach. In: Lecture Notes in Computational Science and Engineering, vol. 6. Springer, Berlin (1999)

  23. Varnhorn, W.: Time stepping procedures for the non-stationary Stokes equations. Math. Methods Appl. Sci. 15, 39–55 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  25. Yan, Y.: Smoothing properties and approximation of time derivatives for parabolic equations: constant time steps. IMA J. Numer. Anal. 23, 465–487 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/A. Springer, New York (1990)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the referees for their comments, which resulted in substantial modifications of the first version of this paper.

Author information

Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, 52056 , Aachen, Germany

    Arnold Reusken & Patrick Esser

Authors
  1. Arnold Reusken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick Esser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnold Reusken.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reusken, A., Esser, P. Analysis of time discretization methods for Stokes equations with a nonsmooth forcing term. Numer. Math. 126, 293–319 (2014). https://doi.org/10.1007/s00211-013-0564-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-013-0564-2

Mathematics Subject Classification (2010)

Search

Navigation