Advertisement

Advances in Computational Mathematics

, Volume 40, Issue 5–6, pp 1093–1119 | Cite as

An adaptive residual local projection finite element method for the Navier–Stokes equations

  • Rodolfo Araya
  • Abner H. PozaEmail author
  • Frédéric Valentin
Article

Abstract

This work proposes and analyses an adaptive finite element scheme for the fully non-linear incompressible Navier-Stokes equations. A residual a posteriori error estimator is shown to be effective and reliable. The error estimator relies on a Residual Local Projection (RELP) finite element method for which we prove well-posedness under mild conditions. Several well-established numerical tests assess the theoretical results.

Keywords

Navier–Stokes equations Stabilized finite element methods A posteriori error estimates 

Mathematics Subject Classifications (2010)

65N12 65N15 65N30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ainsworth, M., Oden, J.: A posteriori error estimators for the Stokes and Oseen equations. SIAM. J. Numer. Anal 34, 228–245 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ainsworth, M., Oden, J.: A posteriori error estimation in finite element analysis. Pur. Appl. Math.Wiley-Interscience, New York (2000)CrossRefGoogle Scholar
  3. 3.
    Araya, R., Barrenechea, G., Poza, A.: An adaptive stabilized finite element method for the generalized Stokes problem. J. Comput. Appl. Math 214(2), 457–479 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Araya, R., Barrenechea, G., Poza, A., Valentin, F.: Convergence analysis of a residual local projection finite element method for the Navier-Stokes equations. SIAM J. Numer. Anal. 50(2), 669–699 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Araya, R., Poza, A., Valentin, F.: On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations. Numer. Methods Partial Differ. Equat. 28(3), 782–806 (2012)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Arnica, D., Padra, C.: A posteriori error estimators for the steady incompressible Navier–Stokes equations. Numer. Methods Partial Differ. Equat. 13(5), 561–574 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Babuska, I., Rheinboldt, W.: Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15(4), 736–754 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bank, R., Welfert, B.: A posteriori error estimators for the Stokes problem. SIAM J. Numer. Anal. 28, 591–623 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Barrenechea, G., Valentin, F.: Consistent local projection stabilized finite element methods. SIAM J. Numer. Anal. 48(5), 1801–1825 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Barrenechea, G., Valentin, F.: A residual local projection method for the Oseen equation. Comput. Methods Appl. Mech. Eng. 199(29–32), 1906–1921 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Becker, R.: An optimal-control approach to a posteriori error estimation for finite element discretizations of the Navier–Stokes equations. East-West. J. Numer. Math. 8(4), 257–274 (2000)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Berrone, S.: Adaptive discretization of stationary and incompressible Navier–Stokes equations by stabilized finite element methods. Comput. Methods Appl. Mech. Eng. 190(34), 4435–4455 (2001)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Braack, M., Burman, E., John, V., Lube, G.: Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Eng. 196(4-6), 853–866 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Clément, P.: Approximation by finite element functions using local regularization. R.A.I.R.O Anal. Numer. 9, 77–84 (1975)zbMATHGoogle Scholar
  15. 15.
    Ern, A., Guermond, J.: Theory and practice of finite elements. In: Applied Mathematical Sciences, Vol. 159. Springer-Verlag, New York (2004)Google Scholar
  16. 16.
    Ervin, V., Layton, W., Maubach, J.: A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations. Numer. Methods Partial Differ. Equat. 12(3), 333–346 (1998)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Franca, L., Frey, S.: Stabilized finite element methods. II. The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 99(2–3), 209–233 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ghia, U., Ghia, K., Shin, C.: High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)CrossRefzbMATHGoogle Scholar
  19. 19.
    Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations. In: Springer Series in Computational Mathematics, Vol. 5. Springer, Berlin (1986)Google Scholar
  20. 20.
    Jin, H., Prudhomme, S.: A posteriori error estimation of steady-state finite element solutions of the Navier–Stokes equations by a subdomain residual method. Comput. Methods Appl. Mech. Eng. 159(1–2), 19–48 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    John, V.: Residual a posteriori error estimates for two-level finite element methods for the Navier–Stokes equations. Appl. Numer. Math. 37(4), 503–518 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Medic, G., Mohammadi, B.: NSIKE – An incompressible Navier–Stokes solver for unstructured meshes. Tech, Vol. 3644. INRIA, Rocquencourt (1999)Google Scholar
  23. 23.
    Oden, J., Wu, W., Ainsworth, M.: An a posteriori error estimate for finite element approximations of the Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 111(1–2), 185–202 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Schäfer, M., Turek, S.: Benchmark computations of laminar flow around a cylinder. Flow simulation with high–performance computer II. Notes Fluid Mech. 52, 547–566 (1996)Google Scholar
  25. 25.
    Schlichting, H., Gersten, K.: Boundary-layer Theory. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  26. 26.
    Shewchuk, J.: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, Vol. 1148, pp. 203–222. Springer (1996)Google Scholar
  27. 27.
    Thomée, V.: Galerkin finite element methods for parabolic problems. In: Springer Series in Computational Mathematics. vol. 25, 2nd edn. Springer, Berlin (2006)Google Scholar
  28. 28.
    Tobiska, L., Verfürth, R.: Analysis of a streamline diffusion finite element method for the Stokes and Navier–Stokes equations. SIAM J. Numer. Anal. 33, 107–127 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Turek, S.: Efficient solvers for incompressible flow problems. In: Lecture Notes in Computational Science and Engineering, Vol. 6. Springer, Berlin (1999)Google Scholar
  30. 30.
    Verfürth, R.: A posteriori error estimators for the Stokes problem. Numer. Math. 55, 309–325 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Verfürth, R.: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62(206), 445–475 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Verfürth, R.: A review of a posteriori error estimation and adaptative mesh–refinement techniques. Wiley–Teubner, Stuttgart (1996)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Rodolfo Araya
    • 1
  • Abner H. Poza
    • 2
    Email author
  • Frédéric Valentin
    • 3
  1. 1.CI2MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.Facultad de IngenieríaUniversidad Católica de la Santísima ConcepciónConcepciónChile
  3. 3.Applied Mathematics DepartmentNational Laboratory for Scientific Computing - LNCCPetrópolis - RJBrazil

Personalised recommendations