Advertisement

Journal of Scientific Computing

, Volume 66, Issue 1, pp 275–295 | Cite as

Towards Optimized Schwarz Methods for the Navier–Stokes Equations

  • Eric Blayo
  • David Cherel
  • Antoine Rousseau
Article

Abstract

This paper presents a study of optimized Schwarz domain decomposition methods for Navier–Stokes equations. Once discretized in time, optimal transparent boundary conditions are derived for the resulting Stokes equations, and a series of local approximations for these nonlocal conditions are proposed. Their convergence properties are studied, and numerical simulations are conducted on the test case of the driven cavity with two subdomains. It is shown that conditions involving one or two degrees of freedom can improve the convergence properties of the original algorithm.

Keywords

Domain decomposition Optimized boundary conditions   Navier–Stokes equations Schwarz alternating method 

Notes

Acknowledgments

This work was supported by the ANR through contract ANR-11-MONU-005 (COMODO) and by the French national programme LEFE/INSU (project CoCoA).

References

  1. 1.
    Brakkee, E., Vuik, C., Wesseling, P.: Domain decomposition for the incompressible Navier–Stokes equations: solving subdomain problems accurately and inaccurately. Int. J. Numer. Methods Fluids 26, 1217–1237 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bruneau, C.H., Saad, Y.: The 2D lid-driven cavity problem revisited. Comput. Fluids 35(3), 23–23 (2006)CrossRefGoogle Scholar
  3. 3.
    Dolean, V., Nataf, F., Rapin, G.: Deriving a new domain decomposition method for the Stokes equations using the Smith factorization. Math. Comput. 78, 789–814 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46, 1175–1199 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 139, 629–651 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Erturk, E.: Discussions on driven cavity flow. Int. J. Numer. Methods Fluids 60, 275–294 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gander, M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44, 699–731 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gunzburger, M., Lee, H.: An optimization-based domain decomposition method for the Navier–Stokes equations. SIAM J. Numer. Anal. 37, 1455–1480 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Halpern, L.: Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems. SIAM J. Math. Anal. 22, 1256–1283 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Halpern, L., Schatzman, M.: Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20, 308–353 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hoang, T.: Space–time domain decomposition methods for mixed formulations of flow and transport problems in porous media. Ph.D. thesis, University of Pierre and Marie Curie (Paris VI) (2013)Google Scholar
  12. 12.
    Japhet, C., Nataf, F.: The best interface conditions for domain decomposition methods: absorbing boundary conditions. In: Tourrette, L., Halpern, L. (eds.) Absorbing Boundaries and Layers, Domain Decomposition Methods. Applications to Large Scale Computations, pp. 348–373. Nova Science, Hauppauge (2001)Google Scholar
  13. 13.
    Japhet, C., Nataf, F., Rogier, F.: The optimized order 2 method: application to convection–diffusion problems. Future Gener. Comput. Syst. 18(1), 17–30 (2001)CrossRefMATHGoogle Scholar
  14. 14.
    Kong, F., Ma, Y., Lu, J.: An optimization-based domain decomposition method for numerical simulation of the incompressible Navier–Stokes flows. Numer. Methods PDEs 27, 255–276 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kumar, P.: Purely algebraic domain decomposition methods for the incompressible Navier–Stokes equations. ArXiv e-prints (2011)Google Scholar
  16. 16.
    Lions, P.L.: On the Schwarz alternating method I. In: Glowinski, R., Golub, G., Meurant, G., Périaux, J. (eds.) Proceedings of the First International Conference on Domain Decomposition Methods, Domain Decomposition Methods for Partial Differential Equations, pp. 1–42. SIAM, Philadelphia (1988)Google Scholar
  17. 17.
    Lions, P.L.: On the Schwarz alternating method III: a variant for nonoverlapping subdomains. In: Chan, T., Glowinski, R., Périaux, J., Widlund, O. (eds.) Proceedings of the Third International Conference on Domain Decomposition Methods, Domain Decomposition Methods for Partial Differential Equations, pp. 202–223. SIAM, Philadelphia (1990)Google Scholar
  18. 18.
    Martin, V.: Schwarz waveform relaxation algorithms for the linear viscous equatorial shallow water equations. SIAM J. Sci. Comput. 31, 3595–3625 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Müller, L., Lube, G.: A nonoverlapping domain decomposition method for the nonstationary Navier–Stokes problem. ZAMM J. Appl. Math. Mech. 81, 725–726 (2001)CrossRefMATHGoogle Scholar
  20. 20.
    Nicolaides, R.: Deflation of conjugate gradients with applications to boundary value problems. SIAM J. Numer. Anal. 24, 355–365 (1987)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Otto, F., Lube, G.: A nonoverlapping domain decomposition method for the Oseen equations. Math. Models Methods Appl. Sci. 8, 1091–1117 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pavarino, L., Widlund, O.: Balancing Neumann–Neumann methods for incompressible Stokes equations. Commun. Pure Appl. Math. 55, 302–335 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ronquist, E.: A domain decomposition solver for the steady Navier–Stokes equations. In: Ilin, A., Scott, L. (eds.) Proceedings of ICOSAHOM-95, pp. 469–485. SIAM, Philadelphia (1996)Google Scholar
  24. 24.
    Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286 (1870)Google Scholar
  25. 25.
    Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps. Numer. Math. 126, 741–770 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Strikwerda, J., Scarbnick, C.: A domain decomposition method for incompressible flow. SIAM J. Sci. Comput. 14, 49–67 (1993)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer, Berlin (2005)MATHGoogle Scholar
  28. 28.
    Willems, J.: Robust multilevel methods for general symmetric positive definite operators. SIAM J. Numer. Anal. 52, 103–124 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Xu, X., Chow, C., Lui, S.H.: On non overlapping domain decomposition methods for the incompressible Navier–Stokes equations. ESAIM Math. Mod. Numer. Anal. 39, 1251–1269 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.LJKUniv. Grenoble AlpesGrenobleFrance
  2. 2.LJKCNRSGrenobleFrance
  3. 3.Inria, Institut de Mathématiques et de Modélisation de MontpellierMontpellierFrance

Personalised recommendations