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 and pressure in ), the scheme is first order convergent in the usual finite volume discrete norm and the 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 norm.
Keywords: finite volumes, colocated discretizations, Stokes problem, incompressible flows, analysis
@article{M2AN_2006__40_3_501_0, author = {Eymard, Robert and Herbin, Rapha\`ele and Latch\'e, Jean Claude}, title = {On a stabilized colocated finite volume scheme for the {Stokes} problem}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {501--527}, publisher = {EDP-Sciences}, volume = {40}, number = {3}, year = {2006}, doi = {10.1051/m2an:2006024}, mrnumber = {2245319}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006024/} }
TY - JOUR AU - Eymard, Robert AU - Herbin, Raphaèle AU - Latché, Jean Claude TI - On a stabilized colocated finite volume scheme for the Stokes problem JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 501 EP - 527 VL - 40 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006024/ DO - 10.1051/m2an:2006024 LA - en ID - M2AN_2006__40_3_501_0 ER -
%0 Journal Article %A Eymard, Robert %A Herbin, Raphaèle %A Latché, Jean Claude %T On a stabilized colocated finite volume scheme for the Stokes problem %J ESAIM: Modélisation mathématique et analyse numérique %D 2006 %P 501-527 %V 40 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2006024/ %R 10.1051/m2an:2006024 %G en %F M2AN_2006__40_3_501_0
Eymard, Robert; Herbin, Raphaèle; Latché, Jean Claude. On a stabilized colocated finite volume scheme for the Stokes problem. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 3, pp. 501-527. doi : 10.1051/m2an:2006024. http://archive.numdam.org/articles/10.1051/m2an:2006024/
[1] Provably good mesh generation. J. Comput. Syst. Sci. 48 (1994) 384-409. | Zbl
, and ,[2] A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. | Zbl
and ,[3] A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457-491. | Zbl
and ,[4] 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. | Zbl
and ,[5] Approximation by finite element functions using local regularization. Rev. Fr. Automat. Infor. R-2 (1975) 77-84. | EuDML | Numdam | Zbl
,[6] A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fl. 46 (2004) 183-201. | Zbl
and ,[7] Finite volume methods. Volume VII of Handbook of Numerical Analysis, North Holland (2000) 713-1020. | Zbl
, and ,[8] 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). | MR
, and ,[9] On colocated clustered finite volume schemes for incompressible flow problems (2006) (in preparation).
, and ,[10] A colocated clustered finite volume schemes based on simplices for the 2D Stokes problem (2006) (in preparation).
, , and ,[11] Computational Methods for Fluid Dynamics. Springer, third edition (2002). | MR | Zbl
and ,[12] Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag 5 (1986). | MR | Zbl
and ,[13] Numerical calculation of time dependent viscous incompressible flow with free surface. Phys. Fluids 8 (1965) 2182-2189.
and ,[14] Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10. | Zbl
and ,[15] Equations aux dérivées partielles. Presses de l'Université de Montréal (1965). | Zbl
,[16] Analysis and convergence of the MAC scheme I. The linear problem. SIAM J. Numer. Anal. 29 (1992) 1579-1591. | Zbl
,[17] Analysis and convergence of the MAC scheme II. Navier-Stokes equations. Math. Comput. 65 (1996) 29-44. | Zbl
and ,[18] Enhancement of the momentum interpolation method on non-staggered grids. Int. J. Numer. Meth. Fl. 33 (2000) 1-22. | Zbl
, and ,[19] Comparison of finite-volume numerical methods with staggered and colocated grids. Comput. Fluids 16 (1988) 389-403. | Zbl
, and ,[20] PELICANS: Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN, 2004.
,[21] Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21 (1983) 1525-1532. | Zbl
and ,[22] Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem. Comput. Method. Appl. M. 79 (1990) 71-86. | Zbl
and ,[23] Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695-713. | Numdam | Zbl
,[24] A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715-719. | Numdam | Zbl
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