Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Local projection stabilization of equal order interpolation applied to the Stokes problem

Authors: Sashikumaar Ganesan, Gunar Matthies and Lutz Tobiska
Journal: Math. Comp. 77 (2008), 2039-2060
MSC (2000): Primary 65N12; Secondary 65N30
Published electronically: May 9, 2008
MathSciNet review: 2429873
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The local projection stabilization allows us to circumvent the Babuška-Brezzi condition and to use equal order interpolation for discretizing the Stokes problem. The projection is usually done in a two-level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements. We propose a new local projection stabilization method based on (possibly) enriched finite element spaces and discontinuous projection spaces defined on the same mesh. Optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf-sup condition. Examples are enriched simplicial finite elements and standard quadrilateral/hexahedral elements. The new approach overcomes the problem of an increasing discretization stencil and, thus, is simple to implement in existing computer codes. Numerical tests confirm the theoretical convergence results which are robust with respect to the user-chosen stabilization parameter.

References [Enhancements On Off] (What's this?)

  • 1. T. Apel, Anisotropic finite elements. Local estimates and applications, Advances in Numerical Mathematics, Teubner, Leipzig, 1999. MR 1716824 (2000k:65002)
  • 2. D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equation, Calcolo 21 (1984), 337-344. MR 799997 (86m:65136)
  • 3. W. Bai, The quadrilateral `Mini' finite element for the Stokes problem, Comput. Methods Appl. Mech. Engrg. 143 (1997), 41-47. MR 1442388 (98c:76061)
  • 4. R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo 38 (2001), no. 4, 173-199. MR 1890352 (2002m:65112)
  • 5. -, A two-level stabilization scheme for the Navier-Stokes equations, Numerical mathematics and advanced applications (Berlin) (M. Feistauer, V. Dolejšı, P. Knobloch, and K. Najzar, eds.), Springer-Verlag, 2004, pp. 123-130. MR 2121360 (2005k:65236)
  • 6. P. Bochev and M. Gunzburger, An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equation, SIAM J. Numer. Anal. 42 (2004), no. 3, 1189-1207. MR 2113682 (2005k:65223)
  • 7. M. Braack and E. Burman, Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method, SIAM J. Numer. Anal. 43 (2006), no. 6, 2544-2566. MR 2206447 (2007a:65139)
  • 8. D. Braess and R. Sarazin, An efficient smoother for the Stokes problem, Appl. Numer. Math. 23 (1997), no. 1, 3-19. MR 1438078 (97k:65277)
  • 9. F. Brezzi and M. Fortin, Mixed hybrid finite element methods, Springer-Verlag, 1991. MR 1115205 (92d:65187)
  • 10. F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes problem, Efficient solutions of elliptic systems (W. Hackbusch, ed.), Notes on Numerical Fluid Mechanics, Vieweg, 1984, pp. 11-19. MR 804083 (86j:65147)
  • 11. P. G. Ciarlet, The finite element method for elliptic problems, SIAM, 2002. MR 1930132
  • 12. P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numer. 9 (1975), 77-84. MR 0400739 (53:4569)
  • 13. R. Codina and J. Blasco, A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation, Comput. Methods Appl. Mech. Engrg. 143 (1997), 373-391. MR 1445157 (98j:76086)
  • 14. J. Douglas and J. Wang, An absolutely stabilized finite element method for the Stokes problem, Math. Comp. 52 (1989), 495-508. MR 958871 (89j:65069)
  • 15. L. P. Franca and S. L. Frey, Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992), no. 2/3, 209-233. MR 1186727 (93i:76055)
  • 16. L. P. Franca, S. P. Oliveira, and M. Sarkis, Continuous Q1/Q1 Stokes elements stabilized with non-conforming null edge average velocity functions, Math. Models Meth. Appl. Sci. (M3AS) 17 (2007), no. 3, 439-459. MR 2311926
  • 17. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equation, Springer Series in Computational Mathematics, no. 5, Springer-Verlag, Berlin, 1986. MR 851383 (88b:65129)
  • 18. P. Hansbo and A. Szepessy, A velocity-pressure streamline diffusion method for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 84 (1990), 175-192. MR 1087615 (91k:76109)
  • 19. T. J. R. Hughes, L. P. Franca, and M. Balestra, A new finite element formulation for computational fluid dynamics. V: Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg. 59 (1986), 85-99. MR 868143 (89j:76015d)
  • 20. V. John and G. Matthies, MooNMD--a program package based on mapped finite element methods, Comput. Vis. Sci. 6 (2004), no. 2-3, 163-169. MR 2061275 (2005a:65132)
  • 21. P. Knobloch and L. Tobiska, Stabilization methods of bubble type for the $ Q_1/Q_1$-element applied to the incompressible Navier-Stokes equations, Math. Mod. Numer. Anal. (M2AN) 34 (2000), no. 1, 85-107. MR 1735975 (2001d:65126)
  • 22. G. Matthies, Mapped finite elements on hexahedra. Necessary and sufficient conditions for optimal interpolation errors, Numer. Algorithms 27 (2001), no. 4, 317-327. MR 1880917 (2002j:65115)
  • 23. G. Matthies, P. Skrzypacz, and L. Tobiska, A unified convergence analysis for local projection stabilisations applied to the Oseen problem, M2AN Math. Model. Numer. Anal. 41 (2007), no. 4, 713-742.
  • 24. G. Matthies and L. Tobiska, The inf-sup condition for the mapped $ Q_k-P_{k-1}^{disc}$ element in arbitrary space dimension, Computing 69 (2002), no. 2, 119-139. MR 1954791 (2004g:65156)
  • 25. P. Mons and G. Rogé, L' élément $ Q_1$-bulle/$ Q_1$, Math. Mod. Numer. Anal. (M2AN) 26 (1992), 507-521. MR 1163979 (93b:65174)
  • 26. M. A. Olshanskii and A. Reusken, Grad-Div stabilization for the Stokes equations, Math. Comp. 73 (2004), 1699-1718. MR 2059732 (2005a:65137)
  • 27. F. Schieweck, Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes, Otto-von-Guericke-Universität Magdeburg, Preprint Nr. 2, 2007.
  • 28. L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput. 54 (1990), no. 190, 483-493. MR 1011446 (90j:65021)
  • 29. L. Tobiska and G. Lube, A modified streamline diffusion method for solving the stationary Navier-Stokes equations, Numer. Math. 59 (1991), 13-29. MR 1103751 (92c:65141)
  • 30. L. Tobiska and R. Verfürth, Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 107-127. MR 1377246 (97e:65133)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N30

Retrieve articles in all journals with MSC (2000): 65N12, 65N30

Additional Information

Sashikumaar Ganesan
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany

Gunar Matthies
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany

Lutz Tobiska
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany

Received by editor(s): April 17, 2007
Received by editor(s) in revised form: September 21, 2007
Published electronically: May 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society