A multi-level correction scheme for eigenvalue problems

Lin, Qun; Xie, Hehu

doi:10.1090/S0025-5718-2014-02825-1

A multi-level correction scheme for eigenvalue problems
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by Qun Lin and Hehu Xie PDF

Math. Comp. 84 (2015), 71-88 Request permission

Abstract:

In this paper, a type of multi-level correction scheme is proposed to solve eigenvalue problems by the finite element method. This type of multi-level correction method includes multi correction steps in a sequence of finite element spaces. In each correction step, we only need to solve a source problem on a finer finite element space and an eigenvalue problem on the coarsest finite element space. The accuracy of the eigenpair approximation can be improved after each correction step. This correction scheme can improve the efficiency of solving eigenvalue problems by the finite element method.

References

A. B. Andreev, R. D. Lazarov, and M. R. Racheva, Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems, J. Comput. Appl. Math. 182 (2005), no. 2, 333–349. MR 2147872, DOI 10.1016/j.cam.2004.12.015
I. Babuška and J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297. MR 962210, DOI 10.1090/S0025-5718-1989-0962210-8
I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Françoise Chatelin, Spectral approximation of linear operators, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici; With solutions to exercises by Mario Ahués. MR 716134
Hongtao Chen, Shanghui Jia, and Hehu Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems, Appl. Math. 54 (2009), no. 3, 237–250. MR 2530541, DOI 10.1007/s10492-009-0015-7
Wei Chen and Qun Lin, Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method, Appl. Math. 51 (2006), no. 1, 73–88. MR 2197324, DOI 10.1007/s10492-006-0006-x
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Xiaozhe Hu and Xiaoliang Cheng, Acceleration of a two-grid method for eigenvalue problems, Math. Comp. 80 (2011), no. 275, 1287–1301. MR 2785459, DOI 10.1090/S0025-5718-2011-02458-0
Shanghui Jia, Hehu Xie, Xiaobo Yin, and Shaoqin Gao, Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods, Appl. Math. 54 (2009), no. 1, 1–15. MR 2476018, DOI 10.1007/s10492-009-0001-0
Qun Lin, Some problems concerning approximate solutions of operator equations, Acta Math. Sinica 22 (1979), no. 2, 219–230 (Chinese, with English summary). MR 542459
Qun Lin, Hung-Tsai Huang, and Zi-Cai Li, New expansions of numerical eigenvalues for $-\Delta u=\lambda \rho u$ by nonconforming elements, Math. Comp. 77 (2008), no. 264, 2061–2084. MR 2429874, DOI 10.1090/S0025-5718-08-02098-X
Q. Lin and J. Lin, Finite element methods: Accuracy and improvement, China Sci. Tech. Press, 2005.
Qun Lin and Tao Lü, Asymptotic expansions for finite element eigenvalues and finite element solution, Extrapolation procedures in the finite element method (Bonn, 1983) Bonner Math. Schriften, vol. 158, Univ. Bonn, Bonn, 1984, pp. 1–10. MR 793412
Qun Lin and Hehu Xie, Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method, Appl. Numer. Math. 59 (2009), no. 8, 1884–1893. MR 2536089, DOI 10.1016/j.apnum.2009.01.011
Q. Lin and N. Yan, The construction and analysis of high efficiency finite element methods, Hebei University Publishers, 1995.
Ahmed Naga, Zhimin Zhang, and Aihui Zhou, Enhancing eigenvalue approximation by gradient recovery, SIAM J. Sci. Comput. 28 (2006), no. 4, 1289–1300. MR 2255457, DOI 10.1137/050640588
Milena R. Racheva and Andrey B. Andreev, Superconvergence postprocessing for eigenvalues, Comput. Methods Appl. Math. 2 (2002), no. 2, 171–185. MR 1930846, DOI 10.2478/cmam-2002-0011
Y. Saad, Numerical Methods for Large Eigenvalue Problems-2nd Edition, SIAM, 2011.
Haijun Wu and Zhimin Zhang, Enhancing eigenvalue approximation by gradient recovery on adaptive meshes, IMA J. Numer. Anal. 29 (2009), no. 4, 1008–1022. MR 2557054, DOI 10.1093/imanum/drn050
Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
Jinchao Xu, A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal. 29 (1992), no. 2, 303–319. MR 1154268, DOI 10.1137/0729020
Jinchao Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231–237. MR 1257166, DOI 10.1137/0915016
Jinchao Xu and Aihui Zhou, A two-grid discretization scheme for eigenvalue problems, Math. Comp. 70 (2001), no. 233, 17–25. MR 1677419, DOI 10.1090/S0025-5718-99-01180-1
Jinchao Xu and Aihui Zhou, Local and parallel finite element algorithms for eigenvalue problems, Acta Math. Appl. Sin. Engl. Ser. 18 (2002), no. 2, 185–200. MR 2008551, DOI 10.1007/s102550200018
Aihui Zhou, Multi-level adaptive corrections in finite dimensional approximations, J. Comput. Math. 28 (2010), no. 1, 45–54. MR 2603580, DOI 10.4208/jcm.2009.09-m1003