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Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods. (English) Zbl 1212.65434
Summary: We analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain a full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements $$Q_1^{\mathrm {rot}}$$ and $$EQ_1^{\mathrm {rot}}$$. Using the eigenvalue error expansion, integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations.

##### MSC:
 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 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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##### References:
 [1] I. Babuška, J. F. Osborn: Estimate for the errors in eigenvalue and eigenvector approximation by Galerkin methods with particular attention to the case of multiple eigenvalue. SIAM J. Numer. Anal. 24 (1987), 1249–1276. · Zbl 0701.65042 [2] I. Babuška, J. F. Osborn: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52 (1989), 275–297. · Zbl 0675.65108 [3] 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 [4] F. Brezzi, M. Fortin: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol. 15. Springer, New York, 1991. · Zbl 0788.73002 [5] W. Chen, Q. Lin: Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method. Appl. Math. 51 (2006), 73–88. · Zbl 1164.65489 [6] W. Chen, Q. Lin: Asymptotic expansion and extrapolation for the eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme. Adv. Comput. Math. 27 (2007), 95–106. · Zbl 1122.65106 [7] Z. Chen: Finite Element Methods and Their Applications. Springer, Berlin, 2005. · Zbl 1082.65118 [8] P. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. · Zbl 0383.65058 [9] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer, Berlin, 1986. [10] 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 [11] H. Han: Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984), 223–233. · Zbl 0573.65083 [12] S. Jia, H. Xie, X. Yin, S. Gao: Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods. Numer. Methods Partial Differ. Equations 24 (2008), 435–448. · Zbl 1151.65086 [13] M. Křížek: Conforming finite element approximation of the Stokes problem. Banach Cent. Publ. 24 (1990), 389–396. [14] Q. Lin, H. Huang, Z. Li: New expansion of numerical eigenvalue for-{$$\Delta$$}u = {$$\lambda$$}u by nonconforming elements. Math. Comput. 77 (2008), 2061–2084. · Zbl 1198.65228 [15] Q. Lin, J. Lin: Finite Element Methods: Accuracy and Improvement. China Sci. Press, Beijing, 2006. [16] Q. Lin, T. Lü: Asymptotic expansions for finite element eigenvalues and finite element solution. Bonn. Math. Schrift 158 (1984), 1–10. [17] Q. Lin, N. Yan: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Press, Hebei, 1996. (In Chinese.) [18] Q. Lin, S. Zhang, N. Yan: Extrapolation and defect correction for diffusion equations with boundary integral conditions. Acta Math. Sci. 17 (1997), 405–412. · Zbl 0907.65096 [19] Q. Lin, Q. Zhu: Preprocessing and Postprocessing for the Finite Element Method. Shanghai Sci. Tech. Publishers, Shanghai, 1994. (In Chinese.) [20] B. Mercier, J. Osborn, J. Rappaz, P.-A. Raviat: Eigenvalue approximation by mixed and hybrid method. Math. Comput. 36 (1981), 427–453. · Zbl 0472.65080 [21] R. Rannacher, S. Turek: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equations 8 (1992), 97–111. · Zbl 0742.76051 [22] V. Shaidurov: Multigrid Methods for Finite Elements. Kluwer Academic Publishers, Dordrecht, 1995. · Zbl 0837.65118 [23] J. Wang, X. Ye: Superconvergence of finite element approximations for the Stokes problem by projection methods. SIAM J. Numer. Anal. 39 (2001), 1001–1013. · Zbl 1002.65118 [24] Y. Yang: An Analysis of the Finite Element Method for Eigenvalue Problems. Guizhou People Public Press, Guizhou, 2004. (In Chinese.) [25] X. Ye: Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Partial Differ. Equations 18 (2002), 143–154. · Zbl 1003.65121 [26] 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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