A Subspace Correction Method for Nearly Singular Linear Elasticity Problems

  • E. KarerEmail author
  • J. K. Kraus
  • L. T. Zikatanov
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 91)


The focus of this work is on constructing a robust (uniform in the problem parameters) iterative solution method for the system of linear algebraic equations arising from a nonconforming finite element discretization based on reduced integration.We introduce a specific space decomposition into two overlapping subspaces that serves as a basis for devising a uniformly convergent subspace correction algorithm. We consider the equations of linear elasticity in primal variables. For nearly incompressible materials, i.e., when the Poisson ratio \(\nu\;\mathrm{approaches}\;1/2 \), this problem becomes ill-posed and the resulting discrete problem is nearly singular.


Multigrid Method Space Decomposition Auxiliary Space Linear Elasticity Problem Iterative Solution 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.



The authors gratefully acknowledge the support by the Austrian Academy of Sciences and by the Austrian Science Fund (FWF), Project No. P19170-N18 and by the National Science Foundation NSF-DMS 0810982.


  1. 1.
    S.C. Brenner and R.L. Scott. The Mathematical Theory of Finite Element Methods (Texts in Applied Mathematics). Springer, 3rd edition, December 2007.Google Scholar
  2. 2.
    F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag New York Inc., 1991.Google Scholar
  3. 3.
    R.S. Falk. Nonconforming finite element methods for the equations of linear elasticity. Math. Comp., 57(196):529–550, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    I. Georgiev, J.K. Kraus, and S. Margenov. Multilevel preconditioning of Crouzeix-Raviart 3D pure displacement elasticity problems. In I. Lirkov et al., editors, LSSC, volume 5910 of LNCS, pages 103–110. Springer, 2010.Google Scholar
  5. 5.
    M. Griebel and P. Oswald. On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math., 70(2):163–180, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    P. Hansbo and M.G. Larson. Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. Math. Model. Numer. Anal., 37(1):63–72, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    R. Hiptmair and J. Xu. Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal., 45(6):2483–2509, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    E. Karer and J. K. Kraus. Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge-matrices and two-level convergence. Int. J. Numer. Meth. Engng., 83(5):642–670, 2010.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J.K. Kraus and S.K. Tomar. Algebraic multilevel iteration method for lowest order Raviart-Thomas space and applications. Int. J. Numer. Meth. Engng., 86(10):1175–1196, 2011.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Y.-J. Lee, J. Wu, J. Xu, and L. T. Zikatanov. Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci., 17(11):1937–1963, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Y.-J. Lee, J. Wu, and J. Chen. Robust multigrid method for the planar linear elasticity problems. Numerische Mathematik, 113:473–496, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J.W. Ruge and K. Stüben. Algebraic multigrid (AMG). In S. F. McCormick, editor, Multigrid Methods, volume 3 of Frontiers Appl. Math., pages 73–130, Philadelphia, 1987. SIAM.Google Scholar
  13. 13.
    J. Schöberl. Multigrid methods for a parameter dependent problem in primal variables. Numer. Math., 84(1):97–119, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    J. Xu. Iterative methods by space decomposition and subspace correction. SIAM Review, 34(4):581–613, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Xu. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing, 56: 215–235, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    J. Xu and L. Zikatanov. The method of alternating projections and the method of subspace corrections in Hilbert space. J. Amer. Math. Soc., 15(3):573–597, 2002.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.Penn State University, State CollegeUniversity ParkUSA

Personalised recommendations