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Lower Bounds for Eigenvalues of the Stokes Operator

Published online by Cambridge University Press:  03 June 2015

Jun Hu  and
Yunqing Huang
Jun Hu*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Yunqing Huang*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan 411105, China
*
Corresponding author. URL: http://math.xtu.edu.cn/myphp/math/personal/huangyq/, Email: hujun@math.pku.edu.cn
Email: huangyq@xtu.edu.cn
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Abstract

In this paper, we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator. We check and prove this condition for four nonconforming methods and one conforming method. Hence they produce eigenvalues which are smaller than their exact counterparts.

Keywords

65N3065N1535J25Lower boundeigenvalueStokes operator
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 1 , February 2013 , pp. 1 - 18
Copyright
Copyright © Global-Science Press 2013

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References

[1]Armentano, M.G. and Duran, R.G., Asymptotic lower bounds for eigenvalues by nonconforming finite element methods, ETNA, 17 (2004), pp. 93–101.Google Scholar
[2]Babuska, I. and Osborn, J.E., Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problem, Math. Comput., 52 (1989), pp. 275–297.CrossRefGoogle Scholar
[3]Babuska, I. and Osborn, J. E., Eigenvalue problems, in Handbook of Numerical Analysis,V.II: Finite Element Methods (Part I), Edited by Ciarlet, P.G. and Lions, J.L., 1991, Elsevier.Google Scholar
[4]Bergh, J. and Löfström, J., Interpolation Spaces: An Introduction, Springer-Verlag Berlin Heidelberg, 1976.Google Scholar
[5]Brenner, S.C. and Scott, L.R., The Mathematical Theorey of Finite Element Methods, Springer-Verlag, 1996.Google Scholar
[6]Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
[7]Chatelin, F., Spectral Approximations of Linear Operators, Academic Press, New York, 1983.Google Scholar
[8]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North–Holland, 1978, reprinted as SIAM Classics in Applied Mathematics, 2002.Google Scholar
[9]Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numér., 7 (1973), pp. 33–76.Google Scholar
[10]Hu, J., Analysis for A Kind of Meshless Galerkin Method and the Lower Approximation of Eigenvalues (in Chinese), Master Thesis, Xiangtan University, 2001.Google Scholar
[11]Hu, J., Huang, Y.Q. and Lin, Q., The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods, arXiv:1112.1145v1[math.NA], 2011.Google Scholar
[12]Hu, J., Huang, Y.Q. and Shen, H. M., The lower approximation of eigenvalue by lumped mass finite element methods, J. Comput. Math., 22 (2004), pp. 545–556.Google Scholar
[13]Hu, J. and Shi, Z. C., The best L2 norm error estimate of lower order finite element methods for the fourth order problem, J. Comput. Math., 30 (2012), pp. 449–460.Google Scholar
[14]Li, Y. A., Lower approximation of eigenvalue by the nonconforming finite element method (in Chinese), Math. Numer. Sin., 30 (2008), pp. 195–200.Google Scholar
[15]Lin, Q., Feng Kang’s spirit and the eigenvalue by the nonconforming mixed element, Workshop on Computational Mathematics and Scientific Computing, On the occasion of the 90-th birthday of the late Professor Feng Kang, 2010, Beijing, China.Google Scholar
[16]Lin, Q., Tobiska, L. and Zhou, A., On the superconvergence of nonconforming low order finite elements applied to the Poisson equation, IMA J. Numer. Anal., 25 (2005), pp. 160–181.CrossRefGoogle Scholar
[17]Lin, Q., Xie, H. H., Luo, F. S. and Li, Y., Stokes eigenvalue approximations from below With nonconforming finite element methods (in Chinese), Math. Prac. Theory, 40 (2010), pp. 157–168.Google Scholar
[18]Osborn, J. E., Spectral approximation for compact operators, Math. Comput., 29 (1975), pp. 712–725.CrossRefGoogle Scholar
[19]Rannacher, R., Nonconforming finite element methods for eigenvalue problems in linear plate theory, Numer. Math., 33 (1979), pp. 23–42.Google Scholar
[20]Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element, Numer. Methods PDEs, 8 (1992), pp. 97–111.CrossRefGoogle Scholar
[21]Shi, Z. C. and Wang, M., The Finite Element Method (in Chinese), Science Press, Beijing, 2010.Google Scholar
[22]Strang, G. and Fix, G., An Analysis of the Finite Element Method, Prentice-Hall, 1973.Google Scholar
[23]Yang, Y. D., Lin, F. B. and Zhang, Z. M., N-simplex Crouzeix-Raviart element for second order elliptic/eigenvalue problems, Inter. J. Numer. Anal. Model., 6 (2009), pp. 615–626.Google Scholar
[24]Yang, Y. D., Zhang, Z. M. and Lin, F. B., Eigenvalue approximation from below using non-forming finite elements, Sci. China Math., 53 (2010), pp. 137–150.Google Scholar
[25]Zhang, Z., Yang, Y. and Chen, Z., Eigenvalue approximation from below by Wilson’s elements (in Chinese), Chinese J. Numer. Math. Appl., 29 (2007), pp. 81–84.Google Scholar