Numerical Algorithms

, Volume 61, Issue 3, pp 465–478 | Cite as

A two-level local projection stabilisation on uniformly refined triangular meshes

  • Gunar Matthies
  • Lutz Tobiska
Original Paper


The two-level local projection stabilisation on triangular meshes is based on the refinement of a macro cell into three child cells by connecting the barycentre with the vertices of the macro cell. This refinement technique leads to non-nested meshes with large inner angles and to non-nested finite element spaces. We show that also the red refinement where a triangle is divided into four child cells by connecting the midpoints of the edges can be used. This avoids the above mentioned disadvantages. For the red refinement a local inf-sup condition for the continuous, piecewise polynomial approximation spaces of order less than or equal to r ≥ 2 living on the refined mesh and discontinuous, piecewise polynomial projection spaces of order less than or equal to r − 1 living on the coarser mesh is established. Numerical tests compare the local projection stabilisation resulting from both refinement rules in case of convection-diffusion problems.


Stabilised finite elements Local projection stabilisation 

Mathematics Subject Classifications (2010)

65N12 65N30 65N15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Becker, R., Braack, M.: A two-level stabilization scheme for the Navier-Stokes equations. In: Feistauer M., Dolejší V., Knobloch P., Najzar K. (eds.) Numerical Mathematics and Advanced Applications, pp. 123–130. Springer-Verlag, Berlin (2004)CrossRefGoogle Scholar
  3. 3.
    Braack, M., Burman, E.: Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43(6), 2544–2566 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002)CrossRefGoogle Scholar
  5. 5.
    Ern, A., Guermond, J.-L.: Theory and practice of finite elements. Applied Mathematical Sciences, vol. 159. Springer-Verlag, New York (2004)Google Scholar
  6. 6.
    Ganesan, S., Matthies, G., Tobiska, L.: Local projection stabilization of equal order interpolation applied to the Stokes problem. Math. Comput. 77(264), 2039–2060 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Guermond, J.-L.: Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN 33, 1293–1316 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    He, L., Tobiska, L.: The two-level local projection stabilization as an enriched one-level approach. Adv. Comput. Math. (2011, published online). doi: 10.1017/s10444-011-9188-1 Google Scholar
  9. 9.
    John, V., Matthies, G.: MooNMD—a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)MathSciNetMATHGoogle Scholar
  10. 10.
    Knobloch, P., Tobiska, L.: On the stability of finite element discretizations of convection diffusion reaction equations. IMA J. Numer. Anal. 31, 147–164 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lube, G.: Stabilized FEM for incompressible flow. Critical review and new trends. In: Wesseling P., Onate E., Périaux J. (eds.) European Conference on Computational Fluid Dynamics ECCOMAS CFD 2006, pp. 1–20. TU Delft, The Netherlands (2006)Google Scholar
  12. 12.
    Matthies, G., Skrzypacz, P., Tobiska, L.: A unified convergence analysis for local projection stabilisations applied to the Oseen problem. Math. Model. Numer. Anal. (M2AN) 41(4), 713–742 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Matthies, G., Skrzypacz, P., Tobiska, L.: Stabilization of local projection type applied to convection-diffusion problems with mixed boundary conditions. ETNA 32, 90–105 (2008)MathSciNetMATHGoogle Scholar
  14. 14.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Springer Series in Computational Mathematics, vol. 24. Springer-Verlag, Berlin (2008)Google Scholar
  15. 15.
    Tobiska, L.: Recent results on local projection stabilization for convection-diffusion and flow problems. In: Hegarty A., Kopteva N., O’Riordan E., Stynes M. (eds.) BAIL 2008 – Boundary and Interior Layers, no. 69 in LNCSE, pp. 55–75. Springer (2009)Google Scholar
  16. 16.
    Tobiska, L., Winkel, C.: The two-level local projection type stabilization as an enriched one-level approach. A one-dimensional study. Int. J. Numer. Anal. Model. 7(3), 520–534 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Fachbereich 10 Mathematik und Naturwissenschaften, Institut für MathematikUniversität KasselKasselGermany
  2. 2.Institut für Analysis und NumerikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations