On a stabilized colocated finite volume scheme for the Stokes problem

Eymard, Robert; Herbin, Raphaèle; Latché, Jean Claude

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 3, pp. 501-527.

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 ${\mathrm{H}}^2\left(\Omega \right)$ and pressure in ${\mathrm{H}}^1\left(\Omega \right)$), the scheme is first order convergent in the usual finite volume discrete ${\mathrm{H}}^1$ norm and the ${\mathrm{L}}^2$ 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 ${\mathrm{L}}^2$ norm.

Classification : 65N12, 65N15, 65N30, 76D07, 76M12
Mots clés : finite volumes, colocated discretizations, Stokes problem, incompressible flows, analysis

[1] M. Bern, D. Eppstein and J. Gilbert, Provably good mesh generation. J. Comput. Syst. Sci. 48 (1994) 384-409. | Zbl

[2] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. | Zbl

[3] F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89 (2001) 457-491. | Zbl

[4] 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. | Zbl

[5] Ph. Clément, Approximation by finite element functions using local regularization. Rev. Fr. Automat. Infor. R-2 (1975) 77-84. | EuDML | Numdam | Zbl

[6] C.R. Dohrmann and P.B. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fl. 46 (2004) 183-201. | Zbl

[7] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods. Volume VII of Handbook of Numerical Analysis, North Holland (2000) 713-1020. | Zbl

[8] 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). | MR

[9] R. Eymard, R. Herbin and J.C. Latché, On colocated clustered finite volume schemes for incompressible flow problems (2006) (in preparation).

[10] 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).

[11] J.H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics. Springer, third edition (2002). | MR | Zbl

[12] 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). | MR | Zbl

[13] F.H. Harlow and J.E. Welsh, Numerical calculation of time dependent viscous incompressible flow with free surface. Phys. Fluids 8 (1965) 2182-2189.

[14] N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput. 58 (1992) 1-10. | Zbl

[15] J. Nečas, Equations aux dérivées partielles. Presses de l'Université de Montréal (1965). | Zbl

[16] R.A. Nicolaides, Analysis and convergence of the MAC scheme I. The linear problem. SIAM J. Numer. Anal. 29 (1992) 1579-1591. | Zbl

[17] R.A. Nicolaides and X. Wu, Analysis and convergence of the MAC scheme II. Navier-Stokes equations. Math. Comput. 65 (1996) 29-44. | Zbl

[18] G. Papageorgakopoulos, G. Arampatzis and N.C. Markatos, Enhancement of the momentum interpolation method on non-staggered grids. Int. J. Numer. Meth. Fl. 33 (2000) 1-22. | Zbl

[19] M. Perić, R. Kessler and G. Scheurer, Comparison of finite-volume numerical methods with staggered and colocated grids. Comput. Fluids 16 (1988) 389-403. | Zbl

[20] B. Piar, PELICANS: Un outil d'implémentation de solveurs d'équations aux dérivées partielles. Note Technique 2004/33, IRSN, 2004.

[21] C.M. Rhie and W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21 (1983) 1525-1532. | Zbl

[22] D.J. Silvester and N. Kechkar, 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

[23] R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695-713. | Numdam | Zbl

[24] R. Verfürth, A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN 33 (1999) 715-719. | Numdam | Zbl