Journal of Scientific Computing

, Volume 71, Issue 1, pp 386–413 | Cite as

Convergence to Suitable Weak Solutions for a Finite Element Approximation of the Navier–Stokes Equations with Numerical Subgrid Scale Modeling

  • Santiago Badia
  • Juan Vicente Gutiérrez-Santacreu
Article
  • 230 Downloads

Abstract

In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition.

Keywords

Navier–Stokes equations Suitable weak solutions Stabilized finite element methods, Subgrid scales 

Mathematics Subject Classification

35Q30 65N30 76N10 

Notes

Acknowledgments

The authors are very grateful to Professor Vivette Girault who provided a proof of a particular case of inequality (8).

This article was funded by Secretaría de Estado de Investigación, Desarrollo e Innovación and European Research Council with Grant Number MTM2015-69875-P and Starting Grant No. 258443 respectively.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev, Spaces Volume 140 of Pure and Applied Mathematics (Amsterdam), 2nd edn. Elsevier/Academic Press, Amsterdam (2013)Google Scholar
  2. 2.
    Badia, S.: On stabilized finite element methods based on the Scott-Zhang projector. Circumventing the inf-sup condition for the Stokes problem. Comput. Methods Appl. Mech. Eng. 247–248, 65–72 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Badia, S., Codina, R., Gutiérrez-Santacreu, J.V.: Long-term stability estimates and existence of a global attractor in a finite element approximation of the Navier-Stokes equations with numerical subgrid scale modeling. SIAM J. Numer. Anal. 48(3), 1013–1037 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Badia, S., Gutiérrez-Santacreu, J.V.: Convergence towards weak solutions of the Navier–Stokes equations for a finite element approximation with numerical subgrid-scale modelling. IMA J. Numer. Anal. 34(3), 1193–1221 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bertoluzza, S.: The discrete commutator property of approximation spaces. C. R. Acad. Sci. Paris Sér. I Math. 329(12), 1097–1102 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boris, J.P., Grinstein, F.F., Oran, E.S., Kolbe, R.L.: New insights into large eddy simulation. Fluid Dyn. Res. 10(4–6), 199 (1992)CrossRefGoogle Scholar
  8. 8.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in applied mathematics, vol. 15, 3rd edn. Springer, New York (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Comm. Pure Appl. Math. 35(6), 771–831 (1982)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Codina, R., Blasco, J.: Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Eng. 182, 277–300 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Codina, R., Principe, J., Guasch, O., Badia, S.: Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 196(21–24), 2413–2430 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Colomés, O., Badia, S., Codina, R., Principe, J.: Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows. Comput. Methods Appl. Mech. Eng. 285, 32–63 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dauge, M.: Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations. SIAM J. Math. Anal. 20(1), 74–97 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004)CrossRefMATHGoogle Scholar
  15. 15.
    Girault, V., Raviart, P.-A.: Finite element approximation of the Navier-Stokes equations, Volume 749 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York (1979)CrossRefGoogle Scholar
  16. 16.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). (Theory and algorithms)Google Scholar
  17. 17.
    Grinstein, F.F., Margollin, L.G., Rider, W.J.: Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  18. 18.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
  19. 19.
    Guasch, O., Codina, R.: Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows. Comput. Methods Appl. Mech. Eng. 261–262, 154–166 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Guermond, J.-L.: Stabilization of Galerkin approximations of transport equations by subgrid modeling. ESAIM Math. Model. Numer. Anal. 33(6), 1293–1316 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Guermond, J.-L.: Finite-element-based Faedo-Galerkin weak solutions to the Navier–Stokes equations in the three-dimensional torus are suitable. Journal de Mathématiques Pures et Appliquées 85(3), 451–464 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Guermond, J.-L.: Faedo-Galerkin weak solutions of the Navier–Stokes equations with Dirichlet boundary conditions are suitable. J. Math. Pures Appl. (9) 88(1), 87–106 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Guermond, J.-L.: On the use of the notion of suitable weak solutions in CFD. Int. J. Numer. Methods Fluids 57(9), 1153–1170 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Guermond, J.-L.: The LBB condition in fractional Sobolev spaces and applications. IMA J. Numer. Anal. 29(3), 790–805 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Guermond, J.-L., Pasciak, J.E.: Stability of discrete Stokes operators in fractional Sobolev spaces. J. Math. Fluid Mech. 10(4), 588–610 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hughes, T.J.R., Feijóo, G.R., Mazzei, L., Quincy, J.-B.: The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Eng. 166(1–2), 3–24 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Hughes, T.J.R., Mazzei, L., Jansen, K.E.: Large eddy simulation and the variational multiscale method. Comput. Vis. Sci. 3, 47–59 (2000)CrossRefMATHGoogle Scholar
  29. 29.
    Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397–431 (1976)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lin, F.H.: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51(3), 241–257 (1998)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  33. 33.
    Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968)MATHGoogle Scholar
  34. 34.
    Scheffer, V.: Hausdorff measure and the Navier–Stokes equations. Comm. Math. Phys. 55(2), 97–112 (1977)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Temam, R.: Navier-Stokes Equations. AMS Chelsea Publishing, Providence (2001). (Theory and numerical analysis, Reprint of the 1984 edition)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Santiago Badia
    • 1
    • 2
  • Juan Vicente Gutiérrez-Santacreu
    • 3
  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Centre Internacional de Mètodes Numèrics en EnginyeriaParc Mediterrani de la TecnologiaCastelldefelsSpain
  3. 3.Dpto. de Matemática Aplicada I, E. T. S. I. InformáticaUniversidad de SevillaSevillaSpain

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