Applications of Mathematics

, Volume 62, Issue 3, pp 225–241 | Cite as

A full multigrid method for semilinear elliptic equation

  • Fei Xu
  • Hehu Xie


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.


semilinear elliptic problem full multigrid method multilevel correction finite element method 

MSC 2010

65N30 65N25 65L15 65B99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. A. Adams: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York, 1975.Google Scholar
  2. [2]
    J. H. Bramble: Multigrid Methods. Pitman Research Notes in Mathematics Series 294, John Wiley & Sons, New York, 1993.Google Scholar
  3. [3]
    J. H. Bramble, J. E. Pasciak: New convergence estimates for multigrid algorithms. Math. Comput. 49 (1987), 311–329.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. H. Bramble, X. Zhang: The analysis of multigrid methods, Handbook of Numerical Analysis. Vol. 7. North-Holland, Amsterdam, 2000, pp. 173–415.MATHGoogle Scholar
  5. [5]
    A. Brandt, S. McCormick, J. Ruge: Multigrid methods for differential eigenproblems. SIAM J. Sci. Stat. Comput. 4 (1983), 244–260.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Springer, New York, 1994.Google Scholar
  7. [7]
    P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam, 1978.Google Scholar
  8. [8]
    W. Hackbusch: Multi-Grid Methods and Applications. Springer Series in Computational Mathematics 4, Springer, Berlin, 1985.Google Scholar
  9. [9]
    Y. Huang, Z. Shi, T. Tang, W. Xue: A multilevel successive iteration method for nonlinear elliptic problems. Math. Comput. 73 (2004), 525–539.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    S. Jia, H. Xie, M. Xie, F. Xu: A full multigrid method for nonlinear eigenvalue problems. Sci. China, Math. 59 (2016), 2037–2048.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Q. Lin, H. Xie: A multi-level correction scheme for eigenvalue problems. Math. Comput. 84 (2015), 71–88.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Q. Lin, H. Xie, F. Xu: Multilevel correction adaptive finite element method for semilinear elliptic equation. Appl. Math., Praha 60 (2015), 527–550.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    L. R. Scott, S. Zhang: Higher-dimensional nonnested multigrid methods. Math. Comput. 58 (1992), 457–466.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    V. V. Shaidurov: Multigrid Methods for Finite Elements. Mathematics and Its Applications 318, Kluwer Academic Publishers, Dordrecht, 1995.Google Scholar
  15. [15]
    A. Toselli, O. B. Widlund: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics 34, Springer, Berlin, 2005.Google Scholar
  16. [16]
    H. Xie: A multigrid method for eigenvalue problem. J. Comput. Phys. 274 (2014), 550–561.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    H. Xie: A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34 (2014), 592–608.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    H. Xie: A multigrid method for nonlinear eigenvalue problems. Sci. Sin., Math. 45 (2015), 1193–1204. (In Chinese.)Google Scholar
  19. [19]
    H. Xie, M. Xie: A multigrid method for ground state solution of Bose-Einstein condensates. Commun. Comput. Phys. 19 (2016), 648–662.MathSciNetCrossRefGoogle Scholar
  20. [20]
    J. Xu: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34 (1992), 581–613.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    J. Xu: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994), 231–237.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    J. Xu: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33 (1996), 1759–1777.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina
  2. 2.LSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations