Advertisement

A Simple Preconditioner for the SIPG Discretization of Linear Elasticity Equations

  • B. Ayuso
  • I. Georgiev
  • J. Kraus
  • L. Zikatanov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6046)

Abstract

We deal with the solution of the systems of linear algebraic equations arising from Symmetric Interior Penalty discontinuous Galerkin (SIPG) discretization of linear elasticity problems in primal (displacement) formulation. The main focus of the paper is on constructing a uniform preconditioner which is based on a natural splitting of the space of piecewise linear discontinuous functions. The presented approach has recently been introduced in [2] in the context of designing subspace correction methods for scalar elliptic partial differential equations and is extended here to linear elasticity equations, i.e., a class of vector field problems. Similar to the scalar case the solution of the linear algebraic system corresponding to the SIPG method is reduced to the solution of a problem arising from discretization by nonconforming Crouzeix-Raviart elements plus the solution of a well-conditioned problem on the complementary space.

Keywords

Linear Elasticity Problem Interior Face Galerkin Discretizations Spectral Condition Number Discontinuous Galerkin Discretizations 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ayuso, B., Georgiev, I., Kraus, J., Zikatanov, L.: A Subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations. RICAM-Report, 16-2009, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria (2009)Google Scholar
  2. 2.
    Ayuso de Dios, B., Zikatanov, L.: Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40(1-3), 4–36 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blaheta, R., Margenov, S., Neytcheva, M.: Aggregation-based multilevel preconditioning of non-conforming FEM elasticity problems. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds.) PARA 2004. LNCS, vol. 3732, pp. 847–856. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Brenner, S., Scott, L.: The mathematical theory of finite element methods. Texts in Applied Mathematics, vol. 15. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  5. 5.
    Brenner, S.C., Sung, L.-Y.: Linear finite element methods for planar linear elasticity. Math. Comp. 59(200), 321–338 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Falk, R.S.: Nonconforming finite element methods for the equations of linear elasticity. Math. Comp. 57(196), 529–550 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Georgiev, I., Kraus, J.K., Margenov, S.: Multilevel preconditioning of Crouzeix-Raviart 3D pure displacement elasticity problems. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2009. LNCS, vol. 5910, pp. 100–107. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Hansbo, P., Larson, M.G.: Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. M2AN Math. Model. Numer. Anal. 37(1), 63–72 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kraus, J.K., Margenov, S.: Robust Algebraic Multilevel Methods and Algorithms. Radon Series on Computational and Applied Mathematics, vol. 5. Walter de Gruyter Inc., New York (October 2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • B. Ayuso
    • 1
  • I. Georgiev
    • 2
  • J. Kraus
    • 3
  • L. Zikatanov
    • 4
  1. 1.Centre de Recerca MatemàticaBarcelonaSpain
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of Sciences AcadSofiaBulgaria
  3. 3.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  4. 4.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations