Parallel Preconditioner for Nonconforming Adini Discretization of a Plate Problem on Nonconforming Meshes

  • Leszek Marcinkowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7203)


In this paper we present a domain decomposition parallel preconditioner for a discretization of a plate problem on nonconforming meshes in 2D. The local discretizations are Adini nonconforming plate finite elements. On the interfaces between adjacent subdomains two mortar conditions are imposed. The condition number of the preconditioned problem is almost optimal i.e. it is bounded poly-logarithmically with respect to the mesh parameters.


Domain Decomposition Mesh Parameter Plate Problem Coarse Space Mortar Method 
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.


  1. 1.
    Bernardi, C., Maday, Y., Patera, A.T.: A new nonconforming approach to domain decomposition: the mortar element method. In: Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, vol. XI (Paris, 1989–1991). Pitman Res. Notes Math. Ser., vol. 299, pp. 13–51. Longman Sci. Tech., Harlow (1994)Google Scholar
  2. 2.
    Ben Belgacem, F.: The mortar finite element method with Lagrange multipliers. Numer. Math. 84(2), 173–197 (1999); First published as a technical report in 1994MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ben Belgacem, F., Maday, Y.: The mortar element method for three-dimensional finite elements. RAIRO Modél. Math. Anal. Numér. 31(2), 289–302 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brenner, S.C., Sung, L.Y.: C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brenner, S.C., Wang, K.: Two-level additive Schwarz preconditioners for C 0 interior penalty methods. Numer. Math. 102(2), 231–255 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brenner, S.C., Sung, L.Y.: Multigrid algorithms for C 0 interior penalty methods. SIAM J. Numer. Anal. 44(1), 199–223 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, vol. II, pp. 17–351. North-Holland, Amsterdam (1991)Google Scholar
  8. 8.
    Toselli, A., Widlund, O.: Domain decomposition methods—algorithms and theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)zbMATHGoogle Scholar
  9. 9.
    Achdou, Y., Kuznetsov, Y.A.: Substructuring preconditioners for finite element methods on nonmatching grids. East-West J. Numer. Math. 3(1), 1–28 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Achdou, Y., Maday, Y., Widlund, O.B.: Iterative substructuring preconditioners for mortar element methods in two dimensions. SIAM J. Numer. Anal. 36(2), 551–580 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bjørstad, P.E., Dryja, M., Rahman, T.: Additive Schwarz methods for elliptic mortar finite element problems. Numer. Math. 95(3), 427–457 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Braess, D., Dahmen, W., Wieners, C.: A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37(1), 48–69 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Marcinkowski, L.: The mortar element method with locally nonconforming elements. BIT 39(4), 716–739 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dryja, M.: A Neumann-Neumann algorithm for a mortar discetization of elliptic problems with discontinuous coefficients. Numer. Math. 99, 645–656 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kim, H.H., Lee, C.O.: A preconditioner for the FETI-DP formulation with mortar methods in two dimensions. SIAM J. Numer. Anal. 42(5), 2159–2175 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Marcinkowski, L., Rahman, T.: Neumann-Neumann algorithms for a mortar Crouzeix-Raviart element for 2nd order elliptic problems. BIT 48(3), 607–626 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Xu, X., Li, L., Chen, W.: A multigrid method for the mortar-type Morley element approximation of a plate bending problem. SIAM J. Numer. Anal. 39(5), 1712–1731 (2001/2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Marcinkowski, L.: Domain decomposition methods for mortar finite element discretizations of plate problems. SIAM J. Numer. Anal. 39(4), 1097–1114 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Marcinkowski, L.: A Neumann-Neumann algorithm for a mortar finite element discretization of fourth-order elliptic problems in 2d. Numer. Methods Partial Differential Equations 25(6), 1425–1442 (2009),, doi:10.1002/num.20406 Published online in Wiley InterScience on December 11, 2008Google Scholar
  20. 20.
    Marcinkowski, L.: A balancing Neumann-Neumann method for a mortar finite element discretization of a fourth order elliptic problem. J. Numer. Math. 18(3), 219–234 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Marcinkowski, L.: A preconditioner for a FETI-DP method for mortar element discretization of a 4th order problem in 2d. Electron. Trans. Numer. Anal. 38, 1–16 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Brenner, S.C.: The condition number of the Schur complement in domain decomposition. Numer. Math. 83(2), 187–203 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2008)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leszek Marcinkowski
    • 1
  1. 1.Faculty of Mathematics, Informatics, and MechanicsUniversity of WarsawWarszawaPoland

Personalised recommendations