Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T10:16:46.720Z Has data issue: false hasContentIssue false
  • English
  • Français

On a stabilized colocated Finite Volume scheme for the Stokes problem

Published online by Cambridge University Press:  22 July 2006

Robert Eymard ,
Raphaèle Herbin  and
Jean Claude Latché
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, France. [email protected]
Raphaèle Herbin
Affiliation:
Université de Provence, France. [email protected]
Jean Claude Latché
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN) – Direction de la Prévention des Accidents Majeurs (DPAM), France. [email protected]
Get access

Abstract

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

Keywords

Finite volumescolocated discretizationsStokes problemincompressible flowsanalysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 3 , May 2006 , pp. 501 - 527
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bern, M., Eppstein, D. and Gilbert, J., Provably good mesh generation. J. Comput. Syst. Sci. 48 (1994) 384–409. CrossRef
Bernardi, C. and Girault, V., A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893–1916. CrossRef
Brezzi, F. and Fortin, M., A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457–491. CrossRef
F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations. In Efficient Solution of Elliptic Systems, W. Hackbusch Ed., Notes Num. Fluid Mech. 10 (1984) 11–19.
Ph. Clément, Approximation by finite element functions using local regularization. Rev. Fr. Automat. Infor. R-2 (1975) 77–84.
Dohrmann, C.R. and Bochev, P.B., A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fl. 46 (2004) 183–201. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. Volume VII of Handbook of Numerical Analysis, North Holland (2000) 713–1020.
R. Eymard, R. Herbin and J.C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes, SIAM J. Numer. Anal. (2006) (in press).
R. Eymard, R. Herbin and J.C. Latché, On colocated clustered finite volume schemes for incompressible flow problems (2006) (in preparation).
R. Eymard, R. Herbin, J.C. Latché and B. Piar, A colocated clustered finite volume schemes based on simplices for the 2D Stokes problem (2006) (in preparation).
J.H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics. Springer, third edition (2002).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag 5 (1986).
Harlow, F.H. and Welsh, J.E., Numerical calculation of time dependent viscous incompressible flow with free surface. Phys. Fluids 8 (1965) 2182–2189. CrossRef
Kechkar, N. and Silvester, D., Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1–10. CrossRef
J. Nečas, Equations aux dérivées partielles. Presses de l'Université de Montréal (1965).
Nicolaides, R.A., Analysis and convergence of the MAC scheme I. The linear problem. SIAM J. Numer. Anal. 29 (1992) 1579–1591. CrossRef
Nicolaides, R.A. and Analysis, X. Wu and convergence of the MAC scheme II. Navier-Stokes equations. Math. Comput. 65 (1996) 29–44. CrossRef
Papageorgakopoulos, G., Arampatzis, G. and Markatos, N.C., Enhancement of the momentum interpolation method on non-staggered grids. Int. J. Numer. Meth. Fl. 33 (2000) 1–22. 3.0.CO;2-0>CrossRef
Perić, M., Kessler, R. and Scheurer, G., Comparison of finite-volume numerical methods with staggered and colocated grids. Comput. Fluids 16 (1988) 389–403. CrossRef
B. Piar, PELICANS: Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN, 2004.
Rhie, C.M. and Chow, W.L., Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21 (1983) 1525–1532. CrossRef
Silvester, D.J. and Kechkar, N., Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. Comput. Method. Appl. M. 79 (1990) 71–86. CrossRef
Verfürth, R., Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695–713. CrossRef
Verfürth, R., A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715–719. CrossRef