Applications of Mathematics

, Volume 52, Issue 1, pp 59–94 | Cite as

Stability of a finite element method for 3D exterior stationary Navier-Stokes flows

  • Paul Deuring
Article

Abstract

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary.

Keywords

stationary incompressible Navier-Stokes flows exterior domains stabilized finite element methods stability estimates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. A. Adams: Sobolev Spaces. Academic Press, New York, 1975.MATHGoogle Scholar
  2. [2]
    F. Alouges, J. Laminie, and S. M. Mefire: Exponential meshes and three-dimensional computation of a magnetic field. Numer. Methods Partial Differ. Equations 19 (2003), 592–637.MathSciNetCrossRefGoogle Scholar
  3. [3]
    K. I. Babenko, M. M. Vasil’ev: On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body. J. Appl. Math. Mech. 37 (1973), 651–665.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    S. Bönisch, V. Heuveline, and P. Wittwer: Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech. 7 (2005), 85–107.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods, 2nd edition. Springer-Verlag, New York, 2002.MATHGoogle Scholar
  6. [6]
    F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.MATHGoogle Scholar
  7. [7]
    C.-H. Bruneau: Boundary conditions on artificial frontiers for incompressible and compressible Navier-Stokes equations. M2AN, Math. Model. Numer. Anal. 34 (2000), 303–314.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    C.-H. Bruneau, P. Fabrie: New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result. M2AN, Math. Model. Numer. Anal. 30 (1996), 815–840.MATHMathSciNetGoogle Scholar
  9. [9]
    C. Calgaro, P. Deuring, and D. Jennequin: A preconditioner for generalized saddle point problems: application to 3D stationary Navier-Stokes equations. Numer. Methods Partial Differ. Equations 22 (2006), 1289–1313.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Deuring: Finite element methods for the Stokes system in three-dimensional exterior domains. Math. Methods Appl. Sci. 20 (1997), 245–269.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Deuring: A stable mixed finite element method on truncated exterior domains. M2AN, Math. Model. Numer. Anal. 32 (1998), 283–305.MATHMathSciNetGoogle Scholar
  12. [12]
    P. Deuring: Approximating exterior flows by flows on truncated exterior domains: piecewise polygonial artificial boundaries. In: Elliptic and Parabolic problems. Proceedings of the 4th European Conference, Rolduc and Gaeta, 2001 (J. Bemelmans, ed.). World Scientific, Singapore, 2002, pp. 364–376.Google Scholar
  13. [13]
    P. Deuring: Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: asymptotic behaviour of the second derivatives of the velocity. Commun. Partial Differ. Equations 30 (2005), 987–1020.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Deuring: A finite element method for 3D exterior Oseen flows: error estimates. Submitted.Google Scholar
  15. [15]
    P. Deuring, S. Kračmar: Artificial boundary conditions for the Oseen system in 3D exterior domains. Analysis 20 (2000), 65–90.MATHGoogle Scholar
  16. [16]
    P. Deuring, S. Kračmar: Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: approximation by flows in bounded domains. Math. Nachr. 269–270 (2004), 86–115.CrossRefGoogle Scholar
  17. [17]
    R. Farwig: A variational approach in weighted Sobolev spaces to the operator −Δ + ∂/∂x 1 in exterior domains of ℝ3. Math. Z. 210 (1992), 449–464.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    R. Farwig: The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211 (1992), 409–447.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Feistauer, C. Schwab: On coupled problems for viscous flows in exterior domains. Math. Models Methods Appl. Sci. 8 (1998), 658–684.CrossRefMathSciNetGoogle Scholar
  20. [20]
    M. Feistauer, C. Schwab: Coupled problems for viscous incompressible flow in exterior domains. In: Applied Nonlinear Analysis (A. Sequeira, ed.). Kluwer/Plenum, New York, 1999, pp. 97–116.Google Scholar
  21. [21]
    M. Feistauer, C. Schwab: Coupling of an interior Navier-Stokes problem with an exterior Oseen problem. J. Math. Fluid Mech. 3 (2001), 1–17.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    R. Finn: On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems. Arch. Ration. Mech. Anal. 19 (1965), 363–406.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations Vol. I. Linearized Steady Problems (rev. ed.). Springer-Verlag, New York, 1998.Google Scholar
  24. [24]
    G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems. Springer-Verlag, New York, 1994.MATHGoogle Scholar
  25. [25]
    V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  26. [26]
    C. I. Goldstein: The finite element method with nonuniform mesh sizes for unbounded domains. Math. Comput. 36 (1981), 387–404.MATHCrossRefGoogle Scholar
  27. [27]
    C. I. Goldstein: Multigrid methods for elliptic problems in unbounded domains. SIAM J. Numer. Anal. 30 (1993), 159–183.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    P. Grisvard: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985.MATHGoogle Scholar
  29. [29]
    G. H. Guirguis: On the coupling of boundary integral and finite element methods for the exterior Stokes problem in 3D. SIAM J. Numer. Anal. 24 (1987), 310–322.MATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    G. H. Guirguis, M. D. Gunzburger: On the approximation of the exterior Stokes problem in three dimensions. M2AN, Math. Model. Numer. Anal. 21 (1987), 445–464.MATHMathSciNetGoogle Scholar
  31. [31]
    M. D. Gunzburger: Finite Element Methods for Viscous Incompressible Flows. Academic Press, Boston, 1989.MATHGoogle Scholar
  32. [32]
    L. Halpern, M. Schatzman: Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20 (1989), 308–353.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    Yinnian He: Coupling boundary integral and finite element methods for the Oseen coupled problem. Comput. Math. Appl. 44 (2002), 1413–1429.MATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    J. G. Heywood, R. Rannacher, and S. Turek: Artificial boundaries and flux and pressure conditions for incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22 (1996), 325–352.MATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    S. Kračmar, J. Neustupa: Global existence of weak solutions of a nonsteady variational inequality of the Navier-Stokes type with mixed boundary conditions. In: Proceedings of the International Symposium on Numerical Analysis (ISNA’92). Charles University, Prague, 1993, pp. 156–177.Google Scholar
  36. [36]
    S. Kračmar, J. Neustupa: A weak solvability of a steady variational inequality of the Navier-Stokes type with mixed boundary conditions. Nonlinear Anal., Theory Methods Appl. 47 (2001), 4169–4180.CrossRefMATHGoogle Scholar
  37. [37]
    P. Kučera: Solutions of the Navier-Stokes equations with mixed boundary conditions in a bounded domain. In: Analysis, Numerics and Applications of Differential and Integral Equations. Pitman Research Notes in Mathematics Series 379 (M. Bach, ed.). Addison Wesley, London, 1998, pp. 127–131.Google Scholar
  38. [38]
    P. Kučera: A structure of the set of critical points to the Navier-Stokes equations with mixed boundary conditions. In: Navier-Stokes Equations: Theory and Numerical Methods. Pitman Research Notes in Mathematics Series 388 (R. Salvi, ed.). Addison Wesley, London, 1998, pp. 201–205.Google Scholar
  39. [39]
    P. Kučera, Z. Skalák: Local solutions to the Navier-Stokes equations with mixed boundary conditions. Acta Appl. Math. 54 (1998), 275–288.CrossRefMathSciNetMATHGoogle Scholar
  40. [40]
    S. A. Nazarov, M. Specovius-Neugenbauer: Approximation of exterior problems. Optimal conditions for the Laplacian. Analysis 16 (1996), 305–324.MATHMathSciNetGoogle Scholar
  41. [41]
    S. A. Nazarov, M. Specovius-Neugenbauer: Approximation of exterior boundary value problems for the Stokes system. Asymptotic Anal. 14 (1997), 233–255.Google Scholar
  42. [42]
    S. A. Nazarov, M. Specovius-Neugebauer: Nonlinear artificial boundary conditions with pointwise error estimates for the exterior three dimensional Navier-Stokes problem. Math. Nachr. 252 (2003), 86–105.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    J. Nečas: Les méthodes directes en théorie des équations elliptiques. Masson, Paris, 1967.Google Scholar
  44. [44]
    K. Nishida: Numerical method for Oseen’s linearized equations in three-dimensional exterior domains. J. Comput. Appl. Math. 152 (2003), 405–409.MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    A. Quarteroni, A. Valli: Numerical Approximation of Partial Differential Equations. Springer-Verlag, New York, 1994.MATHGoogle Scholar
  46. [46]
    T. C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems. Numer. Math. 79 (1998), 283–319.MATHCrossRefMathSciNetGoogle Scholar
  47. [47]
    A. Sequeira: The coupling of boundary integral and finite element methods for the bidimensional exterior steady Stokes problem. Math. Methods Appl. Sci. 5 (1983), 356–375.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    A. Sequeira: On the computer implementation of a coupled boundary and finite element method for the bidimensional exterior steady Stokes problem. Math. Methods Appl. Sci. 8 (1986), 117–133.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    S. V. Tsynkov: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27 (1998), 465–532.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • Paul Deuring
    • 1
  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du LittoralCalais cédexFrance

Personalised recommendations