Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

Hongtao Chen; Shanghui Jia; Hehu Xie

Applications of Mathematics (2009)

  • Volume: 54, Issue: 3, page 237-250
  • ISSN: 0862-7940

Abstract

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In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space.

How to cite

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Chen, Hongtao, Jia, Shanghui, and Xie, Hehu. "Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems." Applications of Mathematics 54.3 (2009): 237-250. <http://eudml.org/doc/37818>.

@article{Chen2009,
abstract = {In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space.},
author = {Chen, Hongtao, Jia, Shanghui, Xie, Hehu},
journal = {Applications of Mathematics},
keywords = {Stokes eigenvalue problem; mixed finite element method; Rayleigh quotient formula; postprocessing; Stokes eigenvalue problem; mixed finite element method; Rayleigh quotient formula},
language = {eng},
number = {3},
pages = {237-250},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems},
url = {http://eudml.org/doc/37818},
volume = {54},
year = {2009},
}

TY - JOUR
AU - Chen, Hongtao
AU - Jia, Shanghui
AU - Xie, Hehu
TI - Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems
JO - Applications of Mathematics
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 3
SP - 237
EP - 250
AB - In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space.
LA - eng
KW - Stokes eigenvalue problem; mixed finite element method; Rayleigh quotient formula; postprocessing; Stokes eigenvalue problem; mixed finite element method; Rayleigh quotient formula
UR - http://eudml.org/doc/37818
ER -

References

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  1. Andreev, A. B., Lazarov, R. D., Racheva, M. R., 10.1016/j.cam.2004.12.015, J. Comput. Appl. Math. 182 (2005), 333-349. (2005) Zbl1075.65136MR2147872DOI10.1016/j.cam.2004.12.015
  2. Babuška, I., Osborn, J. E., 10.1090/S0025-5718-1989-0962210-8, Math. Comput. 52 (1989), 275-297. (1989) MR0962210DOI10.1090/S0025-5718-1989-0962210-8
  3. Babuška, I., Osborn, J., 10.1016/S1570-8659(05)80042-0, In: Handbook of Numerical Analysis, Vol. II. Finite Element Methods (Part 1) J.-L. Lions, P. G. Ciarlet North-Holland Amsterdam (1991), 641-787. (1991) MR1115240DOI10.1016/S1570-8659(05)80042-0
  4. Bacuta, C., Bramble, J. H., 10.1007/s00033-003-3211-4, Z. Angew. Math. Phys. (Special issue dedicated to Lawrence E. Payne) 54 (2003), 874-878. (2003) MR2019187DOI10.1007/s00033-003-3211-4
  5. Bacuta, C., Bramble, J. H., Pasciak, J. E., Shift theorems for the biharmonic Dirichlet problem, In: Recent Progress in Computational and Appl. PDEs. Proceedings of the International Symposium on Computational and Applied PDEs, Zhangiajie, China, July 1-7, 2001 Kluwer Academic/Plenum Publishers New York (2001). (2001) MR2039554
  6. Bernardi, C., Raugel, B., 10.1090/S0025-5718-1985-0771031-7, Math. Comput. 44 (1985), 71-79. (1985) MR0771031DOI10.1090/S0025-5718-1985-0771031-7
  7. Blum, H., Rannacher, R., 10.1002/mma.1670020416, Math. Methods Appl. Sci. 2 (1980), 556-581. (1980) Zbl0445.35023MR0595625DOI10.1002/mma.1670020416
  8. Brenner, S. C., Scott, R. L., The Mathematical Theory of Finite Element Methods, Springer New York (1994). (1994) Zbl0804.65101MR1278258
  9. Brezzi, F., Fortin, M., Mixed and Hybrid Finite Element Methods, Springer New York (1991). (1991) Zbl0788.73002MR1115205
  10. Chatelin, F., Spectral Approximation of Linear Operators, Academic Press New York (1983). (1983) Zbl0517.65036MR0716134
  11. Chen, W., Lin, Q., 10.1007/s10492-006-0006-x, Appl. Math. 51 (2006), 73-88. (2006) Zbl1164.65489MR2197324DOI10.1007/s10492-006-0006-x
  12. Ciarlet, P. G., The Finite Element Method for Elliptic Problem, North-Holland Amsterdam (1978). (1978) MR0520174
  13. Fabes, E. B., Kenig, C. E., Verchota, G. C., The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1998), 769-793. (1998) MR0975121
  14. Girault, V., Raviart, P., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Berlin (1986). (1986) Zbl0585.65077MR0851383
  15. Grisvard, P., Singularities in Boundary Problems, Masson and Springer Paris (1985). (1985) 
  16. Křížek, M., 10.4064/-24-1-389-396, Banach Cent. Publ. 24 (1990), 389-396. (1990) MR1097422DOI10.4064/-24-1-389-396
  17. 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
  18. Lin, Q., Lin, J., Finite Element Methods: Accuracy and Improvement, China Sci. Tech. Press Beijing (2005). (2005) 
  19. Lin, Q., Lü, T., Asymptotic expansions for finite element eigenvalues and finite element solution, Bonn. Math. Schr. 158 (1984), 1-10. (1984) MR0793412
  20. Lin, Q., Yan, N., The Construction and Analysis of High Efficiency Finite Element Methods, Hebei University Publishers Baoding (1995). (1995) 
  21. Mercier, B., Osborn, J., Rappaz, J., Raviart, P. A., 10.1090/S0025-5718-1981-0606505-9, Math. Comput. 36 (1981), 427-453. (1981) Zbl0472.65080MR0606505DOI10.1090/S0025-5718-1981-0606505-9
  22. Osborn, J., 10.1137/0713019, SIAM J. Numer. Anal. 13 (1976), 185-197. (1976) Zbl0334.76010MR0447842DOI10.1137/0713019
  23. Racheva, M. R., Andreev, A. B., 10.2478/cmam-2002-0011, Comput. Methods Appl. Math. 2 (2002), 171-185. (2002) Zbl1012.65113MR1930846DOI10.2478/cmam-2002-0011
  24. Wieners, C., 10.1007/BF01781561, Arch. Math. 66 (1996), 420-427. (1996) Zbl0854.65092MR1383907DOI10.1007/BF01781561
  25. Xu, J., Zhou, A., 10.1090/S0025-5718-99-01180-1, Math. Comput. 70 (2001), 17-25. (2001) Zbl0959.65119MR1677419DOI10.1090/S0025-5718-99-01180-1

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