Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 651808, 27 pages
http://dx.doi.org/10.1155/2012/651808
Research Article

Stabilized Multiscale Nonconforming Finite Element Method for the Stationary Navier-Stokes Equations

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

Received 22 May 2012; Accepted 8 August 2012

Academic Editor: Rudong Chen

Copyright © 2012 Tong Zhang 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. V. Girault and P.-A. Raviart, Finite Element Method for Navier-Stokes equations: Theory and Algorithms, vol. 5, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar
  2. R. Codina and J. Blasco, “Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations,” Numerische Mathematik, vol. 87, no. 1, pp. 59–81, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. Y. He, “A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem,” IMA Journal of Numerical Analysis, vol. 23, no. 4, pp. 665–691, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. Li and Z. Chen, “A new local stabilized nonconforming finite element method for the Stokes equations,” Computing, vol. 82, no. 2-3, pp. 157–170, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G. R. Barrenechea and F. Valentin, “An unusual stabilized finite element method for a generalized Stokes problem,” Numerische Mathematik, vol. 92, no. 4, pp. 653–677, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. R. Becker and P. Hansbo, “A simple pressure stabilization method for the Stokes equation,” Communications in Numerical Methods in Engineering with Biomedical Applications, vol. 24, no. 11, pp. 1421–1430, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Bochev and M. Gunzburger, “An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 3, pp. 1189–1207, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. T. Zhang and Y. N. He, “Fully discrete finite element method based on pressure stabilization for the transient Stokes equations,” Mathematics and Computers in Simulation, vol. 82, pp. 1496–1515, 2012. View at Google Scholar
  9. J. Li and Y. He, “A stabilized finite element method based on two local Gauss integrations for the Stokes equations,” Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 58–65, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. J. Li, Y. He, and Z. Chen, “Performance of several stabilized finite element methods for the Stokes equations based on the lowest equal-order pairs,” Computing, vol. 86, no. 1, pp. 37–51, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. Li, Y. He, and Z. Chen, “A new stabilized finite element method for the transient Navier-Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 1–4, pp. 22–35, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. R. Araya, G. R. Barrenechea, and F. Valentin, “A stabilized finite-element method for the Stokes problem including element and edge residuals,” IMA Journal of Numerical Analysis, vol. 27, no. 1, pp. 172–197, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. R. Barrenechea and F. Valentin, “Relationship between multiscale enrichment and stabilized finite element methods for the generalized Stokes problem,” Comptes Rendus Mathématique. Académie des Sciences, vol. 341, no. 10, pp. 635–640, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. He and K. Li, “Two-level stabilized finite element methods for the steady Navier-Stokes problem,” Computing, vol. 74, no. 4, pp. 337–351, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. L. Franca, A. Madureira, and F. Valentin, “Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions,” Computer Methods in Applied Mechanics and Engineering, vol. 194, pp. 2077–2094, 2005. View at Google Scholar
  16. R. Araya, G. R. Barrenechea, and F. Valentin, “Stabilized finite element methods based on multiscaled enrichment for the Stokes problem,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 322–348, 2006. View at Publisher · View at Google Scholar
  17. L. P. Franca, J. V. A. Ramalho, and F. Valentin, “Multiscale finite element methods for unsteady reaction-diffusion problems,” Communications in Numerical Methods in Engineering, vol. 22, no. 6, pp. 619–625, 2006. View at Google Scholar
  18. M. Crouzeix and P. A. Raviart, “Conforming and nonconforming finite element methods for the stationary Stokes equations I,” RAIRO, vol. 7, pp. 33–76, 1973. View at Google Scholar · View at Scopus
  19. L. Zhu, J. Li, and Z. Chen, “A new local stabilized nonconforming finite element method for solving stationary Navier-Stokes equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2821–2831, 2011. View at Publisher · View at Google Scholar
  20. R. Temam, Navier-Stokes Equation: Theory and Numerical Analysis, vol. 2, North-Holland, Amsterdam, The Netherland, 3rd edition, 1984.
  21. Z. X. Chen, Finite Element Methods and Their Applications, Springer, Heidelerg, Germany, 2005.
  22. 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
  23. F. Hecht, O. Pironneau, A. Le Hyaric, and K. Ohtsuka, May 2008, http://www.freefem.org/ff++.
  24. U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” Journal of Computational Physics, vol. 48, no. 3, pp. 387–411, 1982. View at Google Scholar · View at Scopus