Robust Preconditioner for H(curl) Interface Problems

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 78)


In this paper, we construct an auxiliary space preconditioner for Maxwell’s equations with interface, and generalize the HX preconditioner developed in [9] to the problem with strongly discontinuous coefficients. For the H(curl) interface problem, we show that the condition number of the HX preconditioned system is uniformly bounded with respect to the coefficients and meshsize.

Key words

HX preconditioner AMG H(curl) systems, Nédéelec interface 


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The first author was supported in part by NSF DMS-0609727, NSFC-10528102 and Alexander von Humboldt Research Award for Senior US Scientists. The second author would like to thank his postdoctoral advisor Professor Michael Holst for his encouragement and support through NSF Awards 0715146 and 0411723.


  1. 1.
    A. Alonso and A. Valli. Some remarks on the characterization of the space of tangential traces of H(rot;Ω) and the construction of an extension operator. Manuscripta Math., 89 (2):159–178, 1996. ISSN 0025-2611.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D.N. Arnold, R.S. Falk, and R. Winther. Multigrid in H(div) and H(curl). Numer. Math., 85:197–218, 2000.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M.Sh. Birman and M.Z. Solomyak. L 2-theory of the Maxwell operator in arbitrary domains. Russian Math. Surv., 42(6):75–96, 1987.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    P.B. Bochev, J.J. Hu, C.M. Siefert, and R.S. Tuminaro. An algebraic multigrid approach based on a compatible gauge reformulation of Maxwell’s equations. Technical Report SAND2007-1633 J, Sandia National Laboratory, 2007.Google Scholar
  5. 5.
    J.H. Bramble and J. Xu. Some estimates for a weighted L 2 projection. Math. Comput., 56:463–476, 1991.MATHMathSciNetGoogle Scholar
  6. 6.
    Z. Chen, L. Wang, and W. Zheng. An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput., 2007.Google Scholar
  7. 7.
    A.S.B.B. Dhia, C. Hazard, and S. Lohrengel. A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math., 59(6):2028–2044 (electronic), 1999. ISSN 0036-1399.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Hiptmair. Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal., 36(1):204–225, 1998. URL Scholar
  9. 9.
    R. Hiptmair and J. Xu. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal., 45:2483–2509, 2007.CrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Hiptmair. Finite elements in computational electromagnetism. Acta Numer., 237–339, 2002.Google Scholar
  11. 11.
    Q. Hu and J. Zou. A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal., 41(5):1682–1708, 2003. URL Scholar
  12. 12.
    Q. Hu and J. Zou. A weighted Helmholtz decomposition and application to domain decomposition for saddle-point Maxwell systems. Technical Report 2007-15 (355), CUHK, 2007.Google Scholar
  13. 13.
    T.V. Kolev and P.S. Vassilevski. Some experience with a H 1-based auxiliary space AMG for H (curl) problems. Technical Report UCRL-TR-221841, Lawrence Livermore Nat. Lab., 2006.Google Scholar
  14. 14.
    T.V. Kolev and P.S. Vassilevski. Parallel auxiliary space AMG for H (curl) problems. J. Comput. Math., 27(5):604–623, 2009.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    T.V. Kolev, J.E. Pasciak, and P.S. Vassilevski. H(curl) auxiliary mesh preconditioning. Numer. Linear Algebra Appl., 15(5):455–471, 2008. URL Scholar
  16. 16.
    P. Monk. Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, NY, 2003. ISBN 0-19-850888-3. URL 885.001.0001.
  17. 17.
    J.E. Pasciak and J. Zhao. Overlapping Schwarz methods in H(curl) on polyhedral domains. J. Numer. Math., 10(3):221–234, 2002. ISSN 1570-2820.MATHMathSciNetGoogle Scholar
  18. 18.
    S. Reitzinger and J. Schöberl. An algebraic multigrid method for finite element discretizations with edge elements. Numer. Linear Algebra Appl., 9(3):223–238, 2002. URL
  19. 19.
    A. Toselli. Overlapping Schwarz methods for Maxwell’s equations in three dimensions. Numer. Math., V86(4):733–752, 2000. URL Scholar
  20. 20.
    A. Toselli and O. Widlund. Domain Decomposition Methods—Algorithms and Theory, volume 34 of Springer Series in Computational Mathematics. Springer, Berlin, 2005. ISBN 3-540-20696-5.Google Scholar
  21. 21.
    J. Xu and Y. Zhu. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci., 18(1):77 –105, 2008.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    J. Xu. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured meshes. Computing, 56:215–235, 1996.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    L.T. Zikatanov. Two-sided bounds on the convergence rate of two-level methods. Numer. Linear Algebra Appl., 15(5):439–454, 2008.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of MathematicsUniversity of California, San Diego (UCSD)La JollaUSA

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