Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

Shanghui Jia; Hehu Xie; Xiaobo Yin; Shaoqin Gao

Applications of Mathematics (2009)

  • Volume: 54, Issue: 1, page 1-15
  • ISSN: 0862-7940

Abstract

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In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q 1 rot and E Q 1 rot . Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.

How to cite

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Jia, Shanghui, et al. "Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods." Applications of Mathematics 54.1 (2009): 1-15. <http://eudml.org/doc/37804>.

@article{Jia2009,
abstract = {In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, $Q_1^\{\{\rm rot\}\}$ and $EQ_1^\{\{\rm rot\}\}$. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.},
author = {Jia, Shanghui, Xie, Hehu, Yin, Xiaobo, Gao, Shaoqin},
journal = {Applications of Mathematics},
keywords = {Stokes eigenvalue problem; stream function-vorticity-pressure method; asymptotic expansion; extrapolation; a posteriori error estimates; nonconforming finite element methods; convergence; stream function-vorticity-pressure method; asymptotic expansion; Stokes eigenvalue problem; nonconforming finite elements; error expansion; convergence},
language = {eng},
number = {1},
pages = {1-15},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods},
url = {http://eudml.org/doc/37804},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Jia, Shanghui
AU - Xie, Hehu
AU - Yin, Xiaobo
AU - Gao, Shaoqin
TI - Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 1
EP - 15
AB - In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, $Q_1^{{\rm rot}}$ and $EQ_1^{{\rm rot}}$. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.
LA - eng
KW - Stokes eigenvalue problem; stream function-vorticity-pressure method; asymptotic expansion; extrapolation; a posteriori error estimates; nonconforming finite element methods; convergence; stream function-vorticity-pressure method; asymptotic expansion; Stokes eigenvalue problem; nonconforming finite elements; error expansion; convergence
UR - http://eudml.org/doc/37804
ER -

References

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  1. Babuška, I., Osborn, J. F., 10.1137/0724082, SIAM J. Numer. Anal. 24 (1987), 1249-1276. (1987) MR0917451DOI10.1137/0724082
  2. Babuška, I., Osborn, J. F., 10.1090/S0025-5718-1989-0962210-8, Math. Comput. 52 (1989), 275-297. (1989) MR0962210DOI10.1090/S0025-5718-1989-0962210-8
  3. Bercovier, M., Pironneau, O., 10.1007/BF01399555, Numer. Math. 33 (1979), 211-224. (1979) Zbl0423.65058MR0549450DOI10.1007/BF01399555
  4. Brezzi, F., Fortin, M., 10.1007/978-1-4612-3172-1_1, Springer New York (1991). (1991) MR1115205DOI10.1007/978-1-4612-3172-1_1
  5. Chen, W., Lin, Q., 10.1007/s10492-006-0006-x, Appl. Math. 51 (2006), 73-88. (2006) Zbl1164.65489MR2197324DOI10.1007/s10492-006-0006-x
  6. Chen, W., Lin, Q., 10.1007/s10444-007-9031-x, Adv. Comput. Math. 27 (2007), 95-106. (2007) Zbl1122.65106MR2317923DOI10.1007/s10444-007-9031-x
  7. Chen, Z., Finite Element Methods and Their Applications, Springer Berlin (2005). (2005) Zbl1082.65118MR2158541
  8. Ciarlet, P., The Finite Element Method for Elliptic Problems, North-Holland Amsterdam (1978). (1978) Zbl0383.65058MR0520174
  9. Girault, V., Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Berlin (1986). (1986) Zbl0585.65077MR0851383
  10. Glowinski, R., Pironneau, O., On a mixed finite element approximation of the Stokes problem. I. Convergence of the approximate solution, Numer. Math. 33 (1979), 397-424. (1979) MR0553350
  11. Han, H., Nonconforming elements in the mixed finite element method, J. Comput. Math. 2 (1984), 223-233. (1984) Zbl0573.65083MR0815417
  12. Jia, S., Xie, H., Yin, X., Gao, S., 10.1002/num.20268, Numer. Methods Partial Differ. Equations 24 (2008), 435-448. (2008) Zbl1151.65086MR2382790DOI10.1002/num.20268
  13. Křížek, M., 10.4064/-24-1-389-396, Banach Cent. Publ. 24 (1990), 389-396. (1990) MR1097422DOI10.4064/-24-1-389-396
  14. Lin, Q., Huang, H., Li, Z., 10.1090/S0025-5718-08-02098-X, Math. Comput. 77 (2008), 2061-2084. (2008) MR2429874DOI10.1090/S0025-5718-08-02098-X
  15. Lin, Q., Lin, J., Finite Element Methods: Accuracy and Improvement, China Sci. Press Beijing (2006). (2006) 
  16. Lin, Q., Lü, T., Asymptotic expansions for finite element eigenvalues and finite element solution, Bonn. Math. Schrift 158 (1984), 1-10. (1984) MR0793412
  17. Lin, Q., Yan, N., The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Press Hebei (1996), Chinese. (1996) 
  18. Lin, Q., Zhang, S., Yan, N., 10.1016/S0252-9602(17)30859-7, Acta Math. Sci. 17 (1997), 405-412. (1997) Zbl0907.65096MR1613231DOI10.1016/S0252-9602(17)30859-7
  19. Lin, Q., Zhu, Q., Preprocessing and Postprocessing for the Finite Element Method, Shanghai Sci. Tech. Publishers Shanghai (1994), Chinese. (1994) 
  20. Mercier, B., Osborn, J., Rappaz, J., Raviat, P.-A., 10.1090/S0025-5718-1981-0606505-9, Math. Comput. 36 (1981), 427-453. (1981) MR0606505DOI10.1090/S0025-5718-1981-0606505-9
  21. Rannacher, R., Turek, S., 10.1002/num.1690080202, Numer. Methods Partial Differ. Equations 8 (1992), 97-111. (1992) Zbl0742.76051MR1148797DOI10.1002/num.1690080202
  22. Shaidurov, V., Multigrid Methods for Finite Elements, Kluwer Academic Publishers Dordrecht (1995). (1995) Zbl0837.65118MR1335921
  23. Wang, J., Ye, X., 10.1137/S003614290037589X, SIAM J. Numer. Anal. 39 (2001), 1001-1013. (2001) Zbl1002.65118MR1860454DOI10.1137/S003614290037589X
  24. Yang, Y., An Analysis of the Finite Element Method for Eigenvalue Problems, Guizhou People Public Press Guizhou (2004), Chinese. (2004) 
  25. Ye, X., 10.1002/num.1036, Numer. Methods Partial Differ. Equations 18 (2002), 143-154. (2002) Zbl1003.65121MR1902289DOI10.1002/num.1036
  26. Zhou, A., Li, J., 10.1007/s002110050070, Numer. Math. 68 (1994), 427-435. (1994) Zbl0823.65110MR1313153DOI10.1007/s002110050070

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