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A Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

Part of: Acceleration of convergence Partial differential equations, boundary value problems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  05 August 2015

Xiaole Han ,
Yu Li  and
Hehu Xie
Xiaole Han
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
Yu Li
Affiliation:
Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China
Hehu Xie*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
*
*Email addresses: hanxiaole@lsec.cc.ac.cn (X. Han), liyu@lsec.cc.ac.cn (Y. Li), hhxie@lsec.cc.ac.cn (H. Xie)
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Abstract

In this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.

Keywords

Steklov eigenvalue problemmultilevel correctionnonconforming finite element method

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N25: Eigenvalue problems 65L15: Eigenvalue problems 65B99: None of the above, but in this section
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 3 , August 2015 , pp. 383 - 405
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ahn, H., Vibration of a pendulum consisiting of a bob suspended from a wire, Quart. Appl. Math., 39 (1981), pp. 109–117.CrossRefGoogle Scholar
[2]Andreev, A. B., Lazarov, R. D. and Racheva, M. R., Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems, J. Comput. Appl. Math., 182 (2005), pp. 333–349.CrossRefGoogle Scholar
[3]Babuška, I. and Osborn, J. E., Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), pp. 275–297.CrossRefGoogle Scholar
[4]Babuška, I. and Osborn, J., Eigenvalue Problems, In Handbook of Numerical Analysis, Vol. II, (Eds. Lions, P. G. and Ciarlet, P.G.), Finite Element Methods (Part 1), North-Holland, Amsterdam, pp. 641–787, 1991.Google Scholar
[5]Bergman, S. and Schiffer, M., Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953.Google Scholar
[6]Bermudez, A., Rodriguez, R. and Santamarina, D., A finite element solution of an added mass formulation for coupled fluid-solid vibrations, Numer. Math., 87 (2000), pp. 201–227.CrossRefGoogle Scholar
[7]Bi, H. and Yang, Y., A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem, Appl. Math. Comput., 217 (2011), pp. 9669–9678.Google Scholar
[8]Bi, H. and Yang, Y., Multiscale discretization scheme based on the Rayleigh quotient iterative method for the Steklov eigenvalue problem, Mathematical Problems in Engineering, (2012), Article Number: 487207, Doi: 10.1155/2012/487207.CrossRefGoogle Scholar
[9]Bramble, J. and Osborn, J., Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators, in: Aziz, A., (Ed.), Math. Foundations of the Finite Element Method with Applications to PDE, Academic, New York, pp. 387–408, 1972.Google Scholar
[10]Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, New York: Springer-Verlag, 1994.CrossRefGoogle Scholar
[11]Bucur, D. and Ionescu, I., Asymptotic analysis and scaling of friction parameters, Z. Angew. Math. Phys. (Zamp), 57 (2006), pp. 1–15.Google Scholar
[12]Chatelin, F., Spectral Approximation of Linear Operators, Academic Press Inc, New York, 1983.Google Scholar
[13]Chen, H., Jia, S. and Xie, H., Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems, Appl. Math., 54(3) (2009), pp. 237–250.CrossRefGoogle Scholar
[14]Ciarlet, P., The finite Element Method for Elliptic Problem, North-holland Amsterdam, 1978.CrossRefGoogle Scholar
[15]Conca, C., Planchard, J. and Vanninathanm, M., Fluid and Periodic Structures, John Wiley & Sons, New York, 1995Google Scholar
[16]Crouzeix, M. and Raviart, P., Conforming and nonconforming finite element for solving the stationary Stokes equations, Rairo Anal. Numer., 3 (1973), pp. 33–75.Google Scholar
[17]Grisvard, P., Singularities in Boundary Value Problems, Masson and Springer-Verlag, 1985.Google Scholar
[18]Hinton, D. and Shaw, J., Differential operators with spectral parameter incompletely in the boundary conditions, Funkcialaj Ekvacioj (Serio Internacial), 33, pp. 363–385, 1990.Google Scholar
[19]Hu, X. and Cheng, X., Acceleration of a two-grid method for eigenvalue problems, Math. Comp., 80(275) (2011), pp. 1287–1301.CrossRefGoogle Scholar
[20]Hu, J., Huang, Y. and Lin, Q., TLower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods, J. Sci. Comput., (2014): Doi: 10.1007/S10915-014-9821-5.CrossRefGoogle Scholar
[21]Li, Q., Lin, Q., Xie, H., Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations, Appl. Math., 58(2) (2013), pp. 129–151.CrossRefGoogle Scholar
[22]Li, Q. and Yang, Y., A two-grid discretization scheme for the Steklov eigenvalue problem, J. Appl. Math. Comput., 36 (2011), pp. 129–139CrossRefGoogle Scholar
[23]Lin, Q. and Xie, H., The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods, Mathematics in Practice and Theory, 42(11) (2012), pp. 219–226.Google Scholar
[24]Lin, Q., Xie, H., Luo, F., Li, Y., and Yang, Y., Stokes eigenvalue approximation from below with nonconforming mixed finite element methods, Math. in Practice and Theory, 19 (2010), pp. 157–168.Google Scholar
[25]Lin, Q. and Lin, J., Finite Element Methods: Accuracy and Improvement, China Sci. Tech. Press, 2005. 1995.Google Scholar
[26]Luo, F., Lin, Q. and Xie, H., Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Sci. China. Math., 55 (2012), pp. 1069–1082.CrossRefGoogle Scholar
[27]Racheva, M. R. and Andreev, A. B., Superconvergence postprocessing for Eigenvalues, Comp. Methods in Appl. Math., 2(2) (2002), pp. 171–185.CrossRefGoogle Scholar
[28]Xu, J., Iterative methods by space decomposition and subspace correction, Siam Review, 34(4) (1992), pp. 581–613.CrossRefGoogle Scholar
[29]Xu, J., A new class of iterative methods for nonselfadjoint or indefinite problems, Siam J. Numer. Anal., 29 (1992), pp. 303–319.CrossRefGoogle Scholar
[30]Xu, J., A novel two-grid method for semilinear elliptic equations, Siam J. Sci. Comput., 15 (1994), 231–237.CrossRefGoogle Scholar
[31]Xu, J. and Zhou, A., A two-grid discretization scheme for eigenvalue problems, Math. Comput., 70(233) (2001), pp. 17–25.CrossRefGoogle Scholar
[32]Xu, J. and Zhou, A., Local and parallel finite element algorithm for eigenvalue problems, Acta Math. Appl. Sin. Engl. Ser., 18(2) (2002), pp. 185–200.Google Scholar
[33]Yang, Y. and Chen, Z., The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators, Science in China Series A, 51(7) (2008), pp. 1232–1242.Google Scholar
[34]Yang, Y. and Bi, H., Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems, 49(20) (2011), Siam J. Numer. Anal., pp. 1602–1624.Google Scholar
[35]Zhou, A., Multilevel adaptive corrections in finite dimensional approximations, J. Comp. Math., 28(1) (2010), pp. 45–54.Google Scholar