Applications of Mathematics, Vol. 62, No. 3, pp. 225-241, 2017


A full multigrid method for semilinear elliptic equation

Fei Xu, Hehu Xie

Received December 12, 2016.  First published May 17, 2017.

Abstract:  A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.
Keywords:  semilinear elliptic problem; full multigrid method; multilevel correction; finite element method
Classification MSC:  65N30, 65N25, 65L15, 65B99
DOI:  10.21136/AM.2017.0344-16


References:
[1] R. A. Adams: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[2] J. H. Bramble: Multigrid Methods. Pitman Research Notes in Mathematics Series 294, John Wiley & Sons, New York (1993). MR 1247694 | Zbl 0786.65094
[3] J. H. Bramble, J. E. Pasciak: New convergence estimates for multigrid algorithms. Math. Comput. 49 (1987), 311-329. DOI 10.2307/2008314 | MR 0906174 | Zbl 0659.65098
[4] J. H. Bramble, X. Zhang: The analysis of multigrid methods. Handbook of Numerical Analysis. Vol. 7 North-Holland, Amsterdam (2000), 173-415. DOI 10.1016/S1570-8659(00)07003-4 | MR 1804746 | Zbl 0972.65103
[5] A. Brandt, S. McCormick, J. Ruge: Multigrid methods for differential eigenproblems. SIAM J. Sci. Stat. Comput. 4 (1983), 244-260. DOI 10.1137/0904019 | MR 0697178 | Zbl 0517.65083
[6] S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Springer, New York (1994). DOI 10.1007/978-1-4757-4338-8 | MR 1278258 | Zbl 0804.65101
[7] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam (1978). DOI 10.1016/s0168-2024(08)x7014-6 | MR 0520174 | Zbl 0383.65058
[8] W. Hackbusch: 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
[9] Y. Huang, Z. Shi, T. Tang, W. Xue: A multilevel successive iteration method for nonlinear elliptic problems. Math. Comput. 73 (2004), 525-539. DOI 10.1090/S0025-5718-03-01566-7 | MR 2028418 | Zbl 1042.65101
[10] S. Jia, H. Xie, M. Xie, F. Xu: A full multigrid method for nonlinear eigenvalue problems. Sci. China, Math. 59 (2016), 2037-2048. DOI 10.1007/s11425-015-0234-x | MR 3549940 | Zbl 1354.65236
[11] Q. Lin, H. Xie: A multi-level correction scheme for eigenvalue problems. Math. Comput. 84 (2015), 71-88. DOI 10.1090/S0025-5718-2014-02825-1 | MR 3266953 | Zbl 1307.65159
[12] Q. Lin, H. Xie, F. Xu: Multilevel correction adaptive finite element method for semilinear elliptic equation. Appl. Math., Praha 60 (2015), 527-550. DOI 10.1007/s10492-015-0110-x | MR 3396479 | Zbl 06486924
[13] L. R. Scott, S. Zhang: Higher-dimensional nonnested multigrid methods. Math. Comput. 58 (1992), 457-466. DOI 10.2307/2153196 | MR 1122077 | Zbl 0772.65077
[14] V. V. Shaidurov: Multigrid Methods for Finite Elements. Mathematics and Its Applications 318, Kluwer Academic Publishers, Dordrecht (1995). DOI 10.1007/978-94-015-8527-9 | MR 1335921 | Zbl 0837.65118
[15] A. Toselli, O. B. Widlund: Domain Decomposition Methods - Algorithms and Theory. Springer Series in Computational Mathematics 34, Springer, Berlin (2005). DOI 10.1007/b137868 | MR 2104179 | Zbl 1069.65138
[16] H. Xie: 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
[17] H. Xie: 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
[18] H. Xie: A multigrid method for nonlinear eigenvalue problems. Sci. Sin., Math. 45 (2015), 1193-1204. (In Chinese). DOI 10.1360/n012014-00187
[19] H. Xie, M. Xie: A multigrid method for ground state solution of Bose-Einstein condensates. Commun. Comput. Phys. 19 (2016), 648-662. DOI 10.4208/cicp.191114.130715a | MR 3480951
[20] J. Xu: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992), 581-613. DOI 10.1137/1034116 | MR 1193013 | Zbl 0788.65037
[21] J. Xu: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994), 231-237. DOI 10.1137/0915016 | MR 1257166 | Zbl 0795.65077
[22] J. Xu: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33 (1996), 1759-1777. DOI 10.1137/S0036142992232949 | MR 1411848 | Zbl 0860.65119

Affiliations:   Fei Xu, Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China, e-mail: xufei@lsec.cc.ac.cn; Hehu Xie, LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhongguancun Donglu, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, e-mail: hhxie@lsec.cc.ac.cn

 
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