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Journal of Applied Mathematics
Volume 2011 (2011), Article ID 129724, 16 pages
http://dx.doi.org/10.1155/2011/129724
Research Article

A Stabilized Mixed Finite Element Method for Single-Phase Compressible Flow

1School of Science, Xi'an Jiaotong University, Xi'an 710049, China
2Center for Computational Geosciences, Xi'an Jiaotong University, Xi'an 710049, China
3Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive, NW Calgary, AB, Canada T2N 1N4

Received 6 December 2010; Accepted 2 January 2011

Academic Editor: Shuyu Sun

Copyright © 2011 Liyun Zhang and Zhangxin Chen. 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. Chen, Finite Element Methods and Their Applications, Scientific Computation, Springer, Berlin, Germany, 2005.
  2. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 1986.
  3. D. N. Arnold, F. Brezzi, and M. Fortin, “A stable finite element for the Stokes equations,” Calcolo, vol. 21, no. 4, pp. 337–344, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J.-C. Nedelec, “Mixed finite elements in R3,” Numerische Mathematik, vol. 35, no. 3, pp. 315–341, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. R. A. Raviart and J. M. Thomas, “A mixed FInite element method for 2nd order elliptic problems,” in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, Eds., vol. 606 of Lecture Notes in Mathematics, pp. 292–315, Springer, New York, NY, USA, 1997. View at Google Scholar
  6. F. Brezzi, J. Douglas, Jr., and L. D. Marini, “Two families of mixed finite elements for second order elliptic problems,” Numerische Mathematik, vol. 47, no. 2, pp. 217–235, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. F. Brezzi, J. Douglas, Jr., R. Durán, and M. Fortin, “Mixed finite elements for second order elliptic problems in three variables,” Numerische Mathematik, vol. 51, no. 2, pp. 237–250, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Z. Chen and J. Douglas, Jr., “Prismatic mixed finite elements for second order elliptic problems,” Calcolo, vol. 26, no. 2–4, pp. 135–148, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. F. Brezzi, M. Fortin, and L. D. Marini, “Mixed finite element methods with continuous stresses,” Mathematical Models & Methods in Applied Sciences, vol. 3, no. 2, pp. 275–287, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. Li and Y. He, “A new stabilized finite element method based on local Gauss integration techniques for the Stokes equations,” Journal of Computational and Applied Mathematics, vol. 214, pp. 58–65, 2008. View at Google Scholar
  11. R. Becker and M. Braack, “A finite element pressure gradient stabilization for the Stokes equations based on local projections,” Calcolo, vol. 38, no. 4, pp. 173–199, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. P. Bochev, C. R. Dohrmann, and M. D. Gunzburger, “Stabilization of low-order mixed finite elements for the Stokes equations,” SIAM Journal on Numerical Analysis, vol. 44, no. 1, pp. 82–101, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. F. Brezzi and M. Fortin, “A minimal stabilisation procedure for mixed finite element methods,” Numerische Mathematik, vol. 89, no. 3, pp. 457–491, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. C. R. Dohrmann and P. Bochev, “A stabilized finite element method for the Stokes problem based on polynomial pressure projections,” International Journal for Numerical Methods in Fluids, vol. 46, no. 2, pp. 183–201, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. 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
  16. D. Silvester, “Stabilized mixed finite element methods,” Numerical Analysis Report 262, 1995. View at Google Scholar
  17. R. Falk, “An analysis of the penalty method and extrapolation for the stationary Stokes equations,” in Advances in Computer Methods for Partial Differential Equations, R. Vichnevetsky, Ed., pp. 66–69, AICA, 1975. View at Google Scholar
  18. T. J. R. Hughes, W. Liu, and A. Brooks, “Finite element analysis of incompressible viscous flows by the penalty function formulation,” Journal of Computational Physics, vol. 30, no. 1, pp. 1–60, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A. Masud and T. J. R. Hughes, “A stabilized mixed finite element method for Darcy flow,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 39-40, pp. 4341–4370, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z. Chen, Z. Wang, and J. Li, “Analysis of the pressure projectionstabilization method for second-order elliptic problems,” to appear.
  21. 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. Archives for Scientific Computing, vol. 86, no. 1, pp. 37–51, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, Pa, USA, 2006.
  23. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978.