Previous |  Up |  Next


eigenvalue problem; Stokes problem; stream function-vorticity-pressure method; asymptotic expansion; extrapolation; a posteriori error estimates
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.
[1] I.  Babuška, J.  Osborn: 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. MR 1115240
[2] M.  Bercovier, O.  Pironneau: Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33 (1979), 211–224. DOI 10.1007/BF01399555 | MR 0549450
[3] P. E. Bjørstad, B. P.  Tjøstheim: High precision solutions of two fourth order eigenvalue problems. Computing 63 (1999), 97–107. DOI 10.1007/s006070050053 | MR 1736662
[4] D.  Boffi, F.  Brezzi, and L.  Gastaldi: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 (2000), 121–140. DOI 10.1090/S0025-5718-99-01072-8 | MR 1642801
[5] D.  Boffi, F.  Brezzi, and L.  Gastaldi: On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 25 (1997), 131–154. MR 1655512
[6] F.  Brezzi, M.  Fortin: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol.  15. Springer-Verlag, New York, 1991. MR 1115205
[7] B. M.  Brown, E. B.  Davies, P. K.  Jimack, and M. D. Mihajlović: A numerical investigation of the solution of a class of fourth-order eigenvalue problems. Proc. R. Soc. Lond. A  456 (2000), 1505–1521. DOI 10.1098/rspa.2000.0573 | MR 1808762
[8] P. G.  Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland Publ., Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[9] P. G.  Ciarlet, P.-A.  Raviart: 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. MR 0657977
[10] V.  Girault, P.-A.  Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin, 1986. MR 0851383
[11] R.  Glowinski, O.  Pironneau: On a mixed finite element approximation of the Stokes problem. I: Convergence of the approximate solution. Numer. Math. 33 (1979), 397–424. DOI 10.1007/BF01399323 | MR 0553350
[12] V.  Heuveline, R.  Rannacher: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001), 107–138. DOI 10.1023/A:1014291224961 | MR 1887731
[13] Q.  Hu, J.  Zou: Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems. Numer. Math. 93 (2002), 333–359. DOI 10.1007/s002110100386 | MR 1941400
[14] K.  Ishihara: A mixed finite element method for the biharmonic eigenvalue problem of plate bending. Publ. Res. Inst. Math. Sci. Kyoto Univ. 14 (1978), 399–414. DOI 10.2977/prims/1195189071 | MR 0509196
[15] M.  Křížek: Comforming finite element approximation of the Stokes problem. Banach Cent. Publ. 24 (1990), 389–396. DOI 10.4064/-24-1-389-396
[16] M.  Křížek, P.  Neittaanmäki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175–198. DOI 10.1007/BF00047538 | MR 0900263
[17] Q.  Lin, J.  Lin: Finite Element Methods: Accuracy and Improvement. China Sci. Tech. Press, Beijing, 2005.
[18] Q.  Lin, T.  Lu: Asymptotic expansions for finite element eigenvalues and finite element solution. Bonn Math. Schr. 158 (1984), 1–10. MR 0793412 | Zbl 0549.65072
[19] Q.  Lin, N.  Yan: High Efficiency FEM Construction and Analysis. Hebei Univ. Press, , 1996.
[20] B.  Mercier, J.  Osborn, J.  Rappaz, and P.-A.  Raviart: Eigenvalue approximation by mixed and hybrid methods. Math. Comput. 36 (1981), 427–453. DOI 10.1090/S0025-5718-1981-0606505-9 | MR 0606505
[21] J.  Osborn: Spectral approximation for compact operators. Math. Comput. 29 (1975), 712–725. DOI 10.1090/S0025-5718-1975-0383117-3 | MR 0383117 | Zbl 0315.35068
[22] J.  Osborn: Approximation of the eigenvalue of a nonselfadjoint operator arising in the study of the stability of stationary solutions of the Navier-Stokes equations. SIAM J. Numer. Anal. 13 (1976), 185–197. DOI 10.1137/0713019 | MR 0447842 | Zbl 0334.76010
[23] R.  Rannacher, S.  Turek: Simple noncomforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111. DOI 10.1002/num.1690080202 | MR 1148797
[24] R.  Rannacher: Noncomforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33 (1979), 23–42. DOI 10.1007/BF01396493 | MR 0545740
[25] R.  Stenberg: Postprocess schemes for some mixed finite elements. RAIRO Modélisation Math. Anal. Numér. 25 (1991), 151–168. DOI 10.1051/m2an/1991250101511 | MR 1086845
[26] R.  Verfürth: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO, Anal. Numér. 18 (1984), 175–182. DOI 10.1051/m2an/1984180201751
[27] J.  Wang, X.  Ye: Superconvergence of finite element approximations for the Stokes problem by the projection methods. SIAM J.  Numer. Anal. 39 (2001), 1001–1013. DOI 10.1137/S003614290037589X | MR 1860454
[28] C.  Wieners: Bounds for the $N$ lowest eigenvalues of fourth-order boundary value problems. Computing 59 (1997), 29–41. DOI 10.1007/BF02684402 | MR 1465309 | Zbl 0883.65082
[29] J.  Xu, A.  Zhou: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (2001), 17–25. DOI 10.1090/S0025-5718-99-01180-1 | MR 1677419
[30] X.  Ye: Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Partial Differ. Equations 18 (2002), 143–154. DOI 10.1002/num.1036 | MR 1902289 | Zbl 1003.65121
[31] A.  Zhou, J.  Li: The full approximation accuracy for the stream function-vorticity-pressure method. Numer. Math. 68 (1994), 427–435. DOI 10.1007/s002110050070 | MR 1313153 | Zbl 0823.65110
Partner of
EuDML logo