On the domain geometry dependence of the LBB condition

Evgenii V. Chizhonkov; Maxim A. Olshanskii

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 5, page 935-951
  • ISSN: 0764-583X

Abstract

top
The LBB condition is well-known to guarantee the stability of a finite element (FE) velocity - pressure pair in incompressible flow calculations. To ensure the condition to be satisfied a certain constant should be positive and mesh-independent. The paper studies the dependence of the LBB condition on the domain geometry. For model domains such as strips and rings the substantial dependence of this constant on geometry aspect ratios is observed. In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those when the LBB condition is violated. In the core of the paper we prove that for any FE velocity-pressure pair satisfying usual approximation hypotheses the mesh-independent limit in the LBB condition is not greater than its continuous counterpart, the constant from the Nečas inequality. For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.

How to cite

top

Chizhonkov, Evgenii V., and Olshanskii, Maxim A.. " On the domain geometry dependence of the LBB condition." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 935-951. <http://eudml.org/doc/197462>.

@article{Chizhonkov2010,
abstract = { The LBB condition is well-known to guarantee the stability of a finite element (FE) velocity - pressure pair in incompressible flow calculations. To ensure the condition to be satisfied a certain constant should be positive and mesh-independent. The paper studies the dependence of the LBB condition on the domain geometry. For model domains such as strips and rings the substantial dependence of this constant on geometry aspect ratios is observed. In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those when the LBB condition is violated. In the core of the paper we prove that for any FE velocity-pressure pair satisfying usual approximation hypotheses the mesh-independent limit in the LBB condition is not greater than its continuous counterpart, the constant from the Nečas inequality. For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments. },
author = {Chizhonkov, Evgenii V., Olshanskii, Maxim A.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {LBB; infsup condition; Stokes problem; domain geometry; eigenvalue problem; finite elements.; domain geometry dependence; LBB condition; finite element velocity-pressure pair; mesh-independent limit; Nečas inequality},
language = {eng},
month = {3},
number = {5},
pages = {935-951},
publisher = {EDP Sciences},
title = { On the domain geometry dependence of the LBB condition},
url = {http://eudml.org/doc/197462},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Chizhonkov, Evgenii V.
AU - Olshanskii, Maxim A.
TI - On the domain geometry dependence of the LBB condition
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 5
SP - 935
EP - 951
AB - The LBB condition is well-known to guarantee the stability of a finite element (FE) velocity - pressure pair in incompressible flow calculations. To ensure the condition to be satisfied a certain constant should be positive and mesh-independent. The paper studies the dependence of the LBB condition on the domain geometry. For model domains such as strips and rings the substantial dependence of this constant on geometry aspect ratios is observed. In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those when the LBB condition is violated. In the core of the paper we prove that for any FE velocity-pressure pair satisfying usual approximation hypotheses the mesh-independent limit in the LBB condition is not greater than its continuous counterpart, the constant from the Nečas inequality. For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.
LA - eng
KW - LBB; infsup condition; Stokes problem; domain geometry; eigenvalue problem; finite elements.; domain geometry dependence; LBB condition; finite element velocity-pressure pair; mesh-independent limit; Nečas inequality
UR - http://eudml.org/doc/197462
ER -

References

top
  1. P.P. Aristov and E.V. Chizhonkov, On the Constant in the LBB condition for rectangular domains. Report No. 9535, Dept. of Math. Univ. of Nijmegen, The Netherlands (1995).  Zbl0871.76051
  2. I. Babuska, The finite element method with Lagrange multipliers. Numer. Math.20 (1973) 179-192.  Zbl0258.65108
  3. D. Boffi, F. Brezzi and L. Gastaldi, On the convergence of eigenvalues for mixed formulations. Ann. Sc. Norm. Sup. Pisa25 (1997) 131-154.  Zbl1003.65052
  4. D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 141-158.  Zbl0938.65126
  5. D. Braess, Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer-Verlag, Berlin, Heidelberg, New York (1997).  
  6. J.H. Bramble and J.E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems. Math. Comp.50 (1988) 1-17.  Zbl0643.65017
  7. F. Brezzi, (1974) On the existence, uniqueness and approximation of the saddle-point problems arising from Lagrange multipliers. Numer. Math.20 (1974) 179-192.  
  8. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Comp. Math.15, Springer-Verlag, New York (1991).  Zbl0788.73002
  9. E.V. Chizhonkov, Application of the Cossera spectrum to the optimization of a method for solving the Stokes Problem. Russ. J. Numer. Anal. Math. Model.9 (1994) 191-199.  Zbl0821.76019
  10. M. Crouzeix, Étude d'une méthode de linéarisation. Résolution des équations de Stokes stationaires. Application aux équations des Navier - Stokes stationaires, Cahiers de l'IRIA (1974) 139-244.  
  11. C.M. Dafermos, Some remarks on Korn's inequality. Z. Angew. Math. Phys.19 (1968) 913-920.  Zbl0169.55904
  12. V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin (1986).  Zbl0585.65077
  13. P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston (1985).  Zbl0695.35060
  14. M. Gunsburger, Finite element methods for viscous incompressible flows. A guide to the theory, practice and algorithms. Academic Press, London (1989).  
  15. C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuska-Aziz. Arch. Ration. Mech. Anal.40 (1971) 384-402.  Zbl0512.73017
  16. G.M. Kobelkov, On equivalent norms in L2. Anal. Math. No. 3 (1977) 177-186.  
  17. U. Langer and W. Queck, On the convergence factor of Uzawa's algorithm. J. Comp. Appl. Math.15 (1986) 191-202.  Zbl0601.76021
  18. S.G. Mikhlin, The spectrum of an operator pencil of the elasticity theory. Uspekhi Mat. Nauk28 (1973) 43-82; English translation in Russian Math. Surveys,28.  
  19. M.A. Olshanskii, Stokes problem with model boundary conditions. Sbornik: Mathematics188 (1997) 603-620.  
  20. M.A. Olshanskii and E.V. Chizhonkov, On the optimal constant in the inf-sup condition for rectangle. Matematicheskie Zametki67 (2000) 387-396.  Zbl0979.76070
  21. B.N. Parlett, The Symmetrical Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, New Jersey (1980).  Zbl0431.65017
  22. R. Rannacher and S. Turek, A simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differential Equation8 (1992) 97-111.  Zbl0742.76051
  23. D. Silvester and A. Wathen, Fast iterative solution of stabilized Stokes systems part II: Using block preconditioners. SIAM J. Numer. Anal.31 (1994) 1352-1367.  Zbl0810.76044
  24. M. Schäfer and S. Turek, Benchmark computations of laminar flow around cylinder, in Flow Simulation with High-Performance Computers II, E.H. Hirschel Ed., Notes on Numerical Fluid Mechanics,52, Vieweg (1996) 547-566.  Zbl0874.76070
  25. G. Strang and G.I. Fix, An analysis of the finite element methods. Prentice-Hall, New-York (1973).  Zbl0278.65116
  26. S. Turek, Efficient solvers for incompressible flow problems: An algorithmic approach in view of computational aspects. LNCSE6, Springer, Heidelberg (1999).  Zbl0930.76002
  27. S. Turek and Chr. Becker, FEATFLOW: Finite element software for the incompressible Navier-Stokes equations: User Manual, Release 1.1. Univ. of Heidelberg (1998) (http://www.featflow.de).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.