A Substructuring Preconditioner for Three-Dimensional Maxwell’s Equations

  • Qiya HuEmail author
  • Shi Shu
  • Jun Zou
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 91)


We propose a new nonoverlapping domain decomposition preconditioner for the discrete system arising from the edge element discretization of the three-dimensional Maxwell’s equations. This preconditioner uses the simplest coarse edge element space induced by the coarse triangulation. We will show that the rate of the PCG convergence with this substructuring preconditioner is quasi-optimal, and is independent of large variations of the coefficients across the local interfaces.


Domain Decomposition Domain Decomposition Method Edge Element Local Solver Iterative Substructuring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



QH was supported by the Major Research Plan of Natural Science Foundation of China G91130015, the Key Project of Natural Science Foundation of China G11031006 and National Basic Research Program of China G2011309702. SS was supported by NSFC Project 91130002 and 11171281, the project of Scientific Research Fund of Hunan Provincial Education Department 10C1265 and 11C1219, and the project of Xiangtan University 10XZX03. JZ was substantially supported by Hong Kong RGC grant (Project 405110) and a Direct Grant from Chinese University of Hong Kong.


  1. 1.
    A. Alonso and A. Valli. Some remarks on the characterization of the space of tangential traces of H(curl; Ω) and the construction of an extension operator. Manuscr. Math., 89: 159–178, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    A. Alonso and A. Valli. An optimal domain decomposition preconditioner for low-frequency time-harmonic maxwell equations. Math. Comp., 68(6):607–631, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    J. Bramble, J. Pasciak, and A. Schatz. The construction of preconditioner for elliptic problems by substructuring. IV. Math. Comp., 53(5):1–24, 1989.MathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Cessenat. Mathematical methods in electromagnetism. World Scientific, River Edge, NJ, 1998.Google Scholar
  5. 5.
    Z. Chen, Q. Du, and J. Zou. Finite element methods with matching and non-matching meshes for maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal., 37:1542–1570, 1999.MathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Ciarlet, Jr., and J. Zou. Fully discrete finite element approaches for time-dependent maxwell’s equations. Numer. Math., 82(8):193–219, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    C. R. Dohrmann and O. Widlund. An iterative substructuring algorithm for two-dimensional problems in H(curl). Technical report, TR2010-936, Courant Institute, New York, 2010.Google Scholar
  8. 8.
    R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237–339, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Q. Hu and J. Zou. A non-overlapping domain decomposition method for maxwell’s equations in three dimensions. SIAM J. Numer. Anal., 41: 1682–1708, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Q. Hu and J. Zou. Substructuring preconditioners for saddle-point problems arising from maxwell’s equations in three dimensions. Math. Comput., 73:35–61, 2004.MathSciNetzbMATHGoogle Scholar
  11. 11.
    P. Monk. Analysis of a finite element method for maxwell’s equations. SIAM J. Numer. Anal., 29:32–56, 1992.MathSciNetCrossRefGoogle Scholar
  12. 12.
    P. Monk. Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford, 2003.zbMATHCrossRefGoogle Scholar
  13. 13.
    J. Nedelec. Mixed finite elements in R 3. Numer. Math., 35:315–341, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. Toselli. Dual-primal FETI algorithms for edge finite-element approximations in 3d. IMA J. Numer. Anal., 26:96–130, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. Toselli and O. Widlund. Domain Decomposition Methods – Algorithms and Theory. Springer, New York, 2004.Google Scholar
  16. 16.
    A. Toselli, O. Widlund, and B. Wohlmuth. An iterative substructuring method for Maxwell’s equations in two dimensions. Math. Comp., 70: 935–949, 2001.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    J. Xu and Y. Zhu. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. M 3 AS, 18:77–105, 2008.Google Scholar
  18. 18.
    J. Xu and J. Zou. Some non-overlapping domain decomposition methods. SIAM Review, 40:857–914, 1998.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.LSEC, Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science, The Chinese Academy of SciencesBeijingChina
  2. 2.School of Mathematics and Computational ScienceXiangtan UniversityHunanChina
  3. 3.Department of MathematicsThe Chinese University of Hong KongShatin, N.T.Hong Kong

Personalised recommendations