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Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method. (English) Zbl 1164.65489
Summary: 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.

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics
35P15 Estimates of eigenvalues in context of PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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[1] I. Babuska, 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.
[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. · Zbl 0423.65058
[3] P. E. Bjorstad, B. P. Tjostheim: High precision solutions of two fourth order eigenvalue problems. Computing 63 (1999), 97–107. · Zbl 0940.65119
[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. · Zbl 0938.65126
[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. · Zbl 1003.65052
[6] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol. 15. Springer-Verlag, New York, 1991. · Zbl 0788.73002
[7] B. M. Brown, E. B. Davies, P. K. Jimack, and M. D. Mihajlovic: A numerical investigation of the solution of a class of fourth-order eigenvalue problems. Proc. R. Soc. Lond. A 456 (2000), 1505–1521. · Zbl 0977.65101
[8] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland Publ., Amsterdam, 1978.
[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.
[10] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin, 1986.
[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. · Zbl 0423.65059
[12] V. Heuveline, R. Rannacher: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001), 107–138. · Zbl 0995.65111
[13] Q. Hu, J. Zou: Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems. Numer. Math. 93 (2002), 333–359. · Zbl 1019.65024
[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. · Zbl 0389.73075
[15] M. Krizek: Comforming finite element approximation of the Stokes problem. Banach Cent. Publ. 24 (1990), 389–396.
[16] M. Krizek, P. Neittaanmaki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175–198. · Zbl 0624.65107
[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. · 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. · Zbl 0472.65080
[21] J. Osborn: Spectral approximation for compact operators. Math. Comput. 29 (1975), 712–725. · 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. · Zbl 0334.76010
[23] R. Rannacher, S. Turek: Simple noncomforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111. · Zbl 0742.76051
[24] R. Rannacher: Noncomforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33 (1979), 23–42. · Zbl 0394.65035
[25] R. Stenberg: Postprocess schemes for some mixed finite elements. RAIRO Modelisation Math. Anal. Numer. 25 (1991), 151–168. · Zbl 0717.65081
[26] R. Verfurth: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO, Anal. Numer. 18 (1984), 175–182.
[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. · Zbl 1002.65118
[28] C. Wieners: Bounds for the N lowest eigenvalues of fourth-order boundary value problems. Computing 59 (1997), 29–41. · Zbl 0883.65082
[29] J. Xu, A. Zhou: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (2001), 17–25. · Zbl 0959.65119
[30] X. Ye: Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Partial Differ. Equations 18 (2002), 143–154. · 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. · Zbl 0823.65110
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