Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method

Wei Chen; Qun Lin

Applications of Mathematics (2006)

  • Volume: 51, Issue: 1, page 73-88
  • ISSN: 0862-7940

Abstract

top
By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.

How to cite

top

Chen, Wei, and Lin, Qun. "Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method." Applications of Mathematics 51.1 (2006): 73-88. <http://eudml.org/doc/33245>.

@article{Chen2006,
abstract = {By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.},
author = {Chen, Wei, Lin, Qun},
journal = {Applications of Mathematics},
keywords = {eigenvalue problem; Stokes problem; stream function-vorticity-pressure method; asymptotic expansion; extrapolation; a posteriori error estimates; eigenvalue problem; Stokes problem; stream function-vorticity-pressure method; error expansion; convergence; eigenfunctions; bilinear finite element; extrapolation; error estimate; numerical experiments},
language = {eng},
number = {1},
pages = {73-88},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method},
url = {http://eudml.org/doc/33245},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Chen, Wei
AU - Lin, Qun
TI - Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 1
SP - 73
EP - 88
AB - By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.
LA - eng
KW - eigenvalue problem; Stokes problem; stream function-vorticity-pressure method; asymptotic expansion; extrapolation; a posteriori error estimates; eigenvalue problem; Stokes problem; stream function-vorticity-pressure method; error expansion; convergence; eigenfunctions; bilinear finite element; extrapolation; error estimate; numerical experiments
UR - http://eudml.org/doc/33245
ER -

References

top
  1. Eigenvalue problems, Handbook of Numerical Analysis, Vol. II, Finite Element Method (Part  I), P. G.  Ciarlet, J. L.  Lions (eds.), North-Holland Publ., Amsterdam, 1991, pp. 641–787. (1991) MR1115240
  2. 10.1007/BF01399555, Numer. Math. 33 (1979), 211–224. (1979) MR0549450DOI10.1007/BF01399555
  3. 10.1007/s006070050053, Computing 63 (1999), 97–107. (1999) MR1736662DOI10.1007/s006070050053
  4. 10.1090/S0025-5718-99-01072-8, Math. Comput. 69 (2000), 121–140. (2000) MR1642801DOI10.1090/S0025-5718-99-01072-8
  5. On the convergence of eigenvalues for mixed formulations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 25 (1997), 131–154. (1997) MR1655512
  6. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol.  15, Springer-Verlag, New York, 1991. (1991) MR1115205
  7. 10.1098/rspa.2000.0573, Proc. R. Soc. Lond. A  456 (2000), 1505–1521. (2000) MR1808762DOI10.1098/rspa.2000.0573
  8. The Finite Element Method for Elliptic Problems, North-Holland Publ., Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  9. A mixed finite element method for the biharmonic equation, Aspects finite Elem. partial Differ. Equat., Proc. Symp. Madison, C.  de  Boor (ed.), Academic Press, New York, 1974, pp. 125–145. (1974) MR0657977
  10. Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer-Verlag, Berlin, 1986. (1986) MR0851383
  11. 10.1007/BF01399323, Numer. Math. 33 (1979), 397–424. (1979) MR0553350DOI10.1007/BF01399323
  12. 10.1023/A:1014291224961, Adv. Comput. Math. 15 (2001), 107–138. (2001) MR1887731DOI10.1023/A:1014291224961
  13. 10.1007/s002110100386, Numer. Math. 93 (2002), 333–359. (2002) MR1941400DOI10.1007/s002110100386
  14. 10.2977/prims/1195189071, Publ. Res. Inst. Math. Sci. Kyoto Univ. 14 (1978), 399–414. (1978) MR0509196DOI10.2977/prims/1195189071
  15. 10.4064/-24-1-389-396, Banach Cent. Publ. 24 (1990), 389–396. (1990) DOI10.4064/-24-1-389-396
  16. 10.1007/BF00047538, Acta Appl. Math. 9 (1987), 175–198. (1987) MR0900263DOI10.1007/BF00047538
  17. Finite Element Methods: Accuracy and Improvement, China Sci. Tech. Press, Beijing, 2005. (2005) 
  18. Asymptotic expansions for finite element eigenvalues and finite element solution, Bonn Math. Schr. 158 (1984), 1–10. (1984) Zbl0549.65072MR0793412
  19. High Efficiency FEM Construction and Analysis, Hebei Univ. Press, , 1996. (1996) 
  20. 10.1090/S0025-5718-1981-0606505-9, Math. Comput. 36 (1981), 427–453. (1981) MR0606505DOI10.1090/S0025-5718-1981-0606505-9
  21. 10.1090/S0025-5718-1975-0383117-3, Math. Comput. 29 (1975), 712–725. (1975) Zbl0315.35068MR0383117DOI10.1090/S0025-5718-1975-0383117-3
  22. 10.1137/0713019, SIAM J. Numer. Anal. 13 (1976), 185–197. (1976) Zbl0334.76010MR0447842DOI10.1137/0713019
  23. 10.1002/num.1690080202, Numer. Methods Partial Differ. Equations 8 (1992), 97–111. (1992) MR1148797DOI10.1002/num.1690080202
  24. 10.1007/BF01396493, Numer. Math. 33 (1979), 23–42. (1979) MR0545740DOI10.1007/BF01396493
  25. 10.1051/m2an/1991250101511, RAIRO Modélisation Math. Anal. Numér. 25 (1991), 151–168. (1991) MR1086845DOI10.1051/m2an/1991250101511
  26. 10.1051/m2an/1984180201751, RAIRO, Anal. Numér. 18 (1984), 175–182. (1984) DOI10.1051/m2an/1984180201751
  27. 10.1137/S003614290037589X, SIAM J.  Numer. Anal. 39 (2001), 1001–1013. (2001) MR1860454DOI10.1137/S003614290037589X
  28. 10.1007/BF02684402, Computing 59 (1997), 29–41. (1997) Zbl0883.65082MR1465309DOI10.1007/BF02684402
  29. 10.1090/S0025-5718-99-01180-1, Math. Comput. 70 (2001), 17–25. (2001) MR1677419DOI10.1090/S0025-5718-99-01180-1
  30. 10.1002/num.1036, Numer. Methods Partial Differ. Equations 18 (2002), 143–154. (2002) Zbl1003.65121MR1902289DOI10.1002/num.1036
  31. 10.1007/s002110050070, Numer. Math. 68 (1994), 427–435. (1994) Zbl0823.65110MR1313153DOI10.1007/s002110050070

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.