About this Journal Submit a Manuscript Table of Contents
Advances in Numerical Analysis
Volume 2012 (2012), Article ID 309871, 14 pages
http://dx.doi.org/10.1155/2012/309871
Research Article

Two-Level Stabilized Finite Volume Methods for Stationary Navier-Stokes Equations

1École Nationale Supérieure des Arts et Métiers-Casablanca, Université Hassan II, B.P. 150, Mohammedia, Morocco
2Department of Mathematics and Computing Sciences, Faculty of Sciences and Technology, University Hassan 1st, B.P. 577, Settat, Morocco

Received 17 December 2011; Accepted 17 February 2012

Academic Editor: Weimin Han

Copyright © 2012 Anas Rachid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. Q. Cai, “On the finite volume element method,” Numerische Mathematik, vol. 58, no. 7, pp. 713–735, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. R. E. Ewing, T. Lin, and Y. Lin, “On the accuracy of the finite volume element method based on piecewise linear polynomials,” SIAM Journal on Numerical Analysis, vol. 39, no. 6, pp. 1865–1888, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. H. Chou and D. Y. Kwak, “Analysis and convergence of a MAC-like scheme for the generalized Stokes problem,” Numerical Methods for Partial Differential Equations, vol. 13, no. 2, pp. 147–162, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. P. Chatzipantelidies, “A finite volume method based on the Crouziex Raviart element for elliptic PDEs in two dimension,” Numerical Mathematics, vol. 82, pp. 409–432, 1999.
  5. R. Ewing, R. Lazarov, and Y. Lin, “Finite volume element approximations of nonlocal reactive flows in porous media,” Numerical Methods for Partial Differential Equations, vol. 16, no. 3, pp. 285–311, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. Guoliang and H. Yinnian, “The finite volume method based on stabilized finite element for the stationary Navier-Stokes problem,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 651–665, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. X. Ye, “A discontinuous finite volume method for the Stokes problems,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 183–198, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S. H. Chou, “Analysis and convergence of a covolume method for the generalized Stokes problem,” Mathematics of Computation, vol. 66, no. 217, pp. 85–104, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. M. Berggren, “A vertex-centered, dual discontinuous Galerkin method,” Journal of Computational and Applied Mathematics, vol. 192, no. 1, pp. 175–181, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. S.-H. Chou and D. Y. Kwak, “Multigrid algorithms for a vertex-centered covolume method for elliptic problems,” Numerische Mathematik, vol. 90, no. 3, pp. 441–458, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. R. Eymard, T. Gallouët, and R. Herbin, “Finite volume methods,” in Handbook of Numerical Analysis, Vol. VII, pp. 713–1020, North-Holland, Amsterdam, The Netherlands, 2000. View at Zentralblatt MATH
  12. V. R. Voller, Basic Control Volume Finite Element Methods for Fluids and Solids, vol. 1, World Scientific, Hackensack, NJ, USA, 2009.
  13. J. Xu, “A novel two-grid method for semilinear elliptic equations,” SIAM Journal on Scientific Computing, vol. 15, no. 1, pp. 231–237, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Xu, “Two-grid discretization techniques for linear and nonlinear PDEs,” SIAM Journal on Numerical Analysis, vol. 33, no. 5, pp. 1759–1777, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. V. Ervin, W. Layton, and J. Maubach, “A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations,” Numerical Methods for Partial Differential Equations, vol. 12, no. 3, pp. 333–346, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. V. Girault and J.-L. Lions, “Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra,” Portugaliae Mathematica, vol. 58, no. 1, pp. 25–57, 2001. View at Zentralblatt MATH
  17. Y. He, “Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 41, no. 4, pp. 1263–1285, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. J. Li, “Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier-Stokes equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1470–1481, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. W. Layton, “A two-level discretization method for the Navier-Stokes equations,” Computers & Mathematics with Applications, vol. 26, no. 2, pp. 33–38, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. W. Layton and W. Lenferink, “Two-level Picard, defect correction for the Navier-Stokes equations,” Applied Mathematics and Computation, vol. 80, pp. 1–12, 1995.
  21. W. Layton and L. Tobiska, “A two-level method with backtracking for the Navier-Stokes equations,” SIAM Journal on Numerical Analysis, vol. 35, no. 5, pp. 2035–2054, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. L. Zhu and Y. He, “Two-level Galerkin-Lagrange multipliers method for the stationary Navier-Stokes equations,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 504–512, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. C. Bi and V. Ginting, “Two-grid finite volume element method for linear and nonlinear elliptic problems,” Numerische Mathematik, vol. 108, no. 2, pp. 177–198, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. C. Chen, M. Yang, and C. Bi, “Two-grid methods for finite volume element approximations of nonlinear parabolic equations,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 123–132, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. C. Chen and W. Liu, “Two-grid finite volume element methods for semilinear parabolic problems,” Applied Numerical Mathematics, vol. 60, no. 1-2, pp. 10–18, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. Chatzipantelidis, R. D. Lazarov, and V. Thomée, “Error estimates for a finite volume element method for parabolic equations in convex polygonal domains,” Numerical Methods for Partial Differential Equations, vol. 20, no. 5, pp. 650–674, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. R. Stenberg, “Analysis of mixed finite elements methods for the Stokes problem: a unified approach,” Mathematics of Computation, vol. 42, no. 165, pp. 9–23, 1984. View at Publisher · View at Google Scholar
  28. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Theory and Algorithms, Springer, Berlin, Germany, 1986.
  29. R. Temam, Navier-Stokes Equations, vol. 2 of Theory and Numerical Analysis, North-Holland, Amsterdam, The Netherlands, 3rd edition, 1984.
  30. X. Ye, “On the relationship between finite volume and finite element methods applied to the Stokes equations,” Numerical Methods for Partial Differential Equations, vol. 17, no. 5, pp. 440–453, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. N. Kechkar and D. Silvester, “Analysis of locally stabilized mixed finite element methods for the Stokes problem,” Mathematics of Computation, vol. 58, no. 197, pp. 1–10, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. G. He, Y. He, and X. Feng, “Finite volume method based on stabilized finite elements for the nonstationary Navier-Stokes problem,” Numerical Methods for Partial Differential Equations, vol. 23, no. 5, pp. 1167–1191, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH