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Keywords:
eigenvalue problem; finite element method; Newton's method; multilevel iteration
Summary:
We propose a new type of multilevel method for solving eigenvalue problems based on Newton's method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.
References:
[1] Babuška, I., Osborn, J. E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52 (1989), 275-297. DOI 10.2307/2008468 | MR 0962210 | Zbl 0675.65108
[2] Babuška, I., Osborn, J.: Eigenvalue problems. Handbook of Numerical Analysis. II: Finite Element Methods (Part 1) North-Holland, Amsterdam P. G. Ciarlet, J. L. Lions (1991), 641-787. MR 1115240 | Zbl 0875.65087
[3] Brandt, A., McCormick, S., Ruge, J.: Multigrid methods for differential eigenproblems. SIAM J. Sci. Stat. Comput. 4 (1983), 244-260. DOI 10.1137/0904019 | MR 0697178 | Zbl 0517.65083
[4] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics 15, Springer, New York (1991). DOI 10.1007/978-1-4612-3172-1 | MR 1115205 | Zbl 0788.73002
[5] Chatelin, F.: Spectral Approximation of Linear Operators. Computer Science and Applied Mathematics, Academic Press, New York (1983). DOI 10.1137/1.9781611970678 | MR 0716134 | Zbl 0517.65036
[6] Davidson, E. R., Thompson, W. J.: Monster matrices: their eigenvalues and eigenvectors. Comput. Phys. 7 (1993), 519-522. DOI 10.1063/1.4823212
[7] Durán, R. G., Padra, C., Rodríguez, R.: A posteriori error estimates for the finite element approximation of eigenvalue problems. Math. Models Methods Appl. Sci. 13 (2003), 1219-1229. DOI 10.1142/S0218202503002878 | MR 1998821 | Zbl 1072.65144
[8] Golub, G. H., Loan, C. F. Van: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore (2013). MR 3024913 | Zbl 1268.65037
[9] Hackbusch, W.: On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method. SIAM J. Numer. Anal. 16 (1979), 201-215. DOI 10.1137/0716015 | MR 0526484 | Zbl 0403.65043
[10] Hackbusch, W.: Multi-Grid Methods and Applications. Springer Series in Computational Mathematics 4, Springer, Berlin (1985). DOI 10.1007/978-3-662-02427-0 | MR 0814495 | Zbl 0595.65106
[11] Kressner, D.: A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114 (2009), 355-372. DOI 10.1007/s00211-009-0259-x | MR 2563153 | Zbl 1191.65054
[12] Larson, M. G.: A posteriori and a priori error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. SIAM J. Numer. Anal. 38 (2000), 608-625. DOI 10.1137/S0036142997320164 | MR 1770064 | Zbl 0974.65100
[13] Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Mathematics Monograph Series 1, Elsevier (2006).
[14] Lin, Q., Xie, H.: An observation on the Aubin-Nitsche lemma and its applications. Math. Pract. Theory 41 Chinese (2011), 247-258. MR 2931490 | Zbl 1265.65235
[15] Lin, Q., Xie, H.: A type of multigrid method for eigenvalue problem. Technical report, Research Report of ICM-SEC, 2011 Available at\ http://www.cc.ac.cn/2011researchreport/201106.pdf\kern0pt
[16] Lin, Q., Xie, H.: A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems. Proc. Int. Conf. Applications of Mathematics 2012 J. Brandts et al. Academy of Sciences of the Czech Republic, Institute of Mathematics, Praha (2012), 134-143. MR 3204407 | Zbl 1313.65298
[17] Lin, Q., Yan, N.: The Construction and Analysis of High Efficiency Finite Element Methods. Hebei University Publishers, Shijiazhuang (1996).
[18] Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Algorithms and Architectures for Advanced Scientific Computing, Manchester University Press, Manchester; Halsted Press, New York (1992). DOI 10.1137/1.9781611970739 | MR 1177405 | Zbl 0991.65039
[19] Shaidurov, V. V.: Multigrid Methods for Finite Elements. Mathematics and Its Applications 318, Kluwer Academic Publishers Group, Dordrecht (1995). DOI 10.1007/978-94-015-8527-9 | MR 1335921 | Zbl 0837.65118
[20] Sleijpen, G. L. G., Vorst, H. A. Van der: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17 (1996), 401-425. DOI 10.1137/S0895479894270427 | MR 1384515 | Zbl 0860.65023
[21] Sleijpen, G. L. G., Vorst, H. A. van der: The Jacobi-Davidson method for eigenvalue problems and its relation with accelerated inexact Newton scheme. IMACS 1996: Iterative Methods in Linear Algebra II S. Margenov, P. Vassilevski Blagoevgrad, Bulgaria (1996).
[22] Sleijpen, G. L. G., Vorst, H. A. Van der: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42 (1998), 267-293. DOI 10.1137/S0036144599363084 | MR 1778354 | Zbl 0949.65028
[23] Xie, H.: A multigrid method for eigenvalue problem. J. Comput. Phys. 274 (2014), 550-561. DOI 10.1016/j.jcp.2014.06.030 | MR 3231782 | Zbl 1352.65631
[24] Xie, H.: A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34 (2014), 592-608. DOI 10.1093/imanum/drt009 | MR 3194801 | Zbl 1312.65178
[25] Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992), 581-613. DOI 10.1137/1034116 | MR 1193013 | Zbl 0788.65037
[26] Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70 (2001), 17-25. DOI 10.1090/S0025-5718-99-01180-1 | MR 1677419 | Zbl 0959.65119
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