Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations

Petr Knobloch; Lutz Tobiska

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 1, page 85-107
  • ISSN: 0764-583X

Abstract

top
In this paper, a general technique is developed to enlarge the velocity space V h 1 of the unstable -element by adding spaces V h 2 such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

How to cite

top

Knobloch, Petr, and Tobiska, Lutz. "Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 85-107. <http://eudml.org/doc/197571>.

@article{Knobloch2010,
abstract = { In this paper, a general technique is developed to enlarge the velocity space $\{\rm V\}_h^1$ of the unstable -element by adding spaces $\{\rm V\}_h^2$ such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case. },
author = {Knobloch, Petr, Tobiska, Lutz},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Babuska-Brezzi condition; stabilization; Stokes equations; Navier-Stokes equations.; stabilized quadilateral element; incompressible Navier-Stokes equations},
language = {eng},
month = {3},
number = {1},
pages = {85-107},
publisher = {EDP Sciences},
title = {Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations},
url = {http://eudml.org/doc/197571},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Knobloch, Petr
AU - Tobiska, Lutz
TI - Stabilization methods of bubble type for the Q1/Q1-element applied to the incompressible Navier-Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 85
EP - 107
AB - In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2/Q1-element and the 4Q1/Q1-element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q1 to the Q2 and 4Q1, respectively, element are not necessary to stabilize the Q1/Q1-element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkäranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.
LA - eng
KW - Babuska-Brezzi condition; stabilization; Stokes equations; Navier-Stokes equations.; stabilized quadilateral element; incompressible Navier-Stokes equations
UR - http://eudml.org/doc/197571
ER -

References

top
  1. D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo21 (1984) 337-344.  Zbl0593.76039
  2. C. Baiocchi, F. Brezzi and L.P. Franca, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.). Comput. Methods Appl. Mech. Eng.105 (1993) 125-141.  Zbl0772.76033
  3. R.E. Bank and B.D Welfert, A comparison between the mini-element and the Petrov-Galerkin formulations for the generalized Stokes problem. Comput. Methods Appl. Mech. Eng. 83 (1990) 61-68.  Zbl0732.65100
  4. M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math.33 (1979) 211-224.  Zbl0423.65058
  5. C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput.44 (1985) 71-79.  Zbl0563.65075
  6. F. Brezzi, M.-O. Bristeau, L.P. Franca, M. Mallet and G. Rogé, A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Eng.96 (1992) 117-129.  Zbl0756.76044
  7. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991).  Zbl0788.73002
  8. F. Brezzi, L.P. Franca and A. Russo, Further considerations on residual-free bubbles for advective-diffusive equations. Technical Report UCD/CCM 113, Univerity of Colorado at Denver, Center for Computational Mathematics (1997).  Zbl0934.65126
  9. F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in Efficient Solutions of Elliptic Systems, W. Hackbusch Ed., Notes on Numerical Fluid Mechanics10 Vieweg-Verlag Braunschweig (1984) 11-19.  
  10. M. Fortin, Old and new finite elements for incompressible flows. Int. J. Numer. Methods Fluids1 (1981) 347-364.  Zbl0467.76030
  11. L.P. Franca and C. Farhat, Bubble functions prompt unusual stabilized finite element methods. Comput. Methods Appl. Mech. Eng. 123 (1995) 299-308.  Zbl1067.76567
  12. L.P. Franca and S.L. Frey, Stabilized finite element methods. II: The incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 99 (1992) 209-233.  Zbl0765.76048
  13. L.P. Franca and A. Russo, Approximation of the Stokes problem by residual-free macro bubbles, East-West J. Numer. Math.4 (1996) 265-278.  Zbl0869.76038
  14. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin (1986).  Zbl0413.65081
  15. T.J. Hughes, L.P. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics. V: Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng.59 (1986) 85-99.  Zbl0622.76077
  16. P. Knobloch, Reduced finite element discretizations of the Stokes and Navier-Stokes equations. J. Math. Fluid Mechanics (to appear).  Zbl1088.76026
  17. P. Mons and G. Rogé, L'élément Q1-bulle/Q1. Math. Mod. Numer. Anal.26 (1992) 507-521.  
  18. R. Pierre, Simple C0 approximations for the computation of incompressible flows. Comput. Methods Appl. Mech. Eng.68 (1988) 205-227.  Zbl0628.76040
  19. T.C. Rebollo, A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math.79 (1998) 283-319.  Zbl0910.76033
  20. A. Russo, Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng.132 (1996) 335-343.  Zbl0887.76038
  21. 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) 107-127.  Zbl0843.76052
  22. R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numér.18 (1984) 175-182.  Zbl0557.76037

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.