Multiplicative Overlapping Schwarz Smoothers for Hdiv-Conforming Discontinuous Galerkin Methods for the Stokes Problem

  • Guido KanschatEmail author
  • Youli Mao
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


We present numerical results for a multigrid method employing overlapping Schwarz smoothers in various V-cycle configurations. The method is based on finite element discretizations of the Stokes problem employing Hdiv-conforming velocity spaces and matching pressure spaces. The method acts on the combined velocity and pressure spaces and thus does not need a Schur complement approximation.


Discontinuous Galerkin methods Divergence-conforming Multigrid Overlapping Schwarz Smoother 


  1. 1.
    D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D.N. Arnold, R.S. Falk, R. Winther, Preconditioning in H(div) and applications. Math. Comput. 66(219), 957–984 (1997). ISSN 0025-5718. doi:10.1090/S0025-5718-97-00826-0Google Scholar
  3. 3.
    W. Bangerth, R. Hartmann, G. Kanschat, deal.II — a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4) (2007). doi:10.1145/1268776.1268779Google Scholar
  4. 4.
    W. Bangerth, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, T.D. Young, The deal.II library, version 8.2. Arch. Numer. Softw. 3(100), 1–8 (2015)Google Scholar
  5. 5.
    B. Cockburn, G. Kanschat, D. Schötzau, C. Schwab, Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002). doi:10.1137/S0036142900380121.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    B. Cockburn, G. Kanschat, D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31(1–2), 61–73 (2007). doi:10.1007/s10915-006-9107-7MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    B. Janssen, G. Kanschat, Adaptive multilevel methods with local smoothing for H 1- and H curl-conforming high order finite element methods. SIAM J. Sci. Comput. 33(4), 2095–2114 (2011) doi:10.1137/090778523.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. Kanschat, Y. Mao, Multigrid methods for H div-conforming discontinuous Galerkin methods for the Stokes equations. J. Numer. Math. (2014, to appear)Google Scholar
  9. 9.
    P.-A. Raviart, J.M. Thomas, A mixed method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, ed. by I. Galligani, E. Magenes (Springer, New York, 1977), pp. 292–315CrossRefGoogle Scholar
  10. 10.
    J. Schöberl, Robust multigrid methods for parameter dependent problems. Dissertation, Johannes Kepler Universität Linz, 1999zbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)Universität HeidelbergHeidelbergGermany
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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