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On a stabilized collocated finite volume scheme for the Stokes problem. (English) Zbl 1160.76370

Summary: We present and analyse in this paper a novel collocated 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 \(H^2(\Omega)\) and pressure in \(H^1(\Omega)\)), the scheme is first order convergent in the usual finite volume discrete \(H^1\) norm and the \(L^2\) 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 \(L^2\) norm.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] M. Bern , D. Eppstein and J. Gilbert , Provably good mesh generation . J. Comput. Syst. Sci. 48 ( 1994 ) 384 - 409 . Zbl 0799.65119 · Zbl 0799.65119 · doi:10.1016/S0022-0000(05)80059-5
[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 0913.65007 · Zbl 0913.65007 · doi:10.1137/S0036142995293766
[3] F. Brezzi and M. Fortin , A minimal stabilisation procedure for mixed finite element methods . Numer. Math. 89 ( 2001 ) 457 - 491 . Zbl 1009.65067 · Zbl 1009.65067 · doi:10.1007/s002110100258
[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 0552.76002 · Zbl 0552.76002
[5] Ph. Clément , Approximation by finite element functions using local regularization . Rev. Fr. Automat. Infor. R- 2 ( 1975 ) 77 - 84 . Numdam | Zbl 0368.65008 · Zbl 0368.65008
[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 1060.76569 · Zbl 1060.76569 · doi:10.1002/fld.752
[7] R. Eymard , T. Gallouët and R. Herbin , Finite volume methods . Volume VII of Handbook of Numerical Analysis, North Holland ( 2000 ) 713 - 1020 . Zbl 0981.65095 · Zbl 0981.65095
[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). arXiv | MR 2285842 | Zbl pre05246514 · Zbl 1173.76028
[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). · Zbl 1137.76062
[11] J.H. Ferziger and M. Perić , Computational Methods for Fluid Dynamics . Springer, third edition ( 2002 ). MR 1745618 | Zbl 0998.76001 · Zbl 0998.76001 · doi:10.1007/978-3-642-56026-2
[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 851383 | Zbl 0585.65077 · Zbl 0585.65077
[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 . · Zbl 1180.76043 · doi:10.1063/1.1761178
[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 0738.76040 · Zbl 0738.76040 · doi:10.2307/2153016
[15] J. Nečas , Equations aux dérivées partielles . Presses de l’Université de Montréal ( 1965 ). Zbl 0147.07801 · Zbl 0147.07801
[16] R.A. Nicolaides , Analysis and convergence of the MAC scheme I . The linear problem. SIAM J. Numer. Anal. 29 ( 1992 ) 1579 - 1591 . Zbl 0764.76051 · Zbl 0764.76051 · doi:10.1137/0729091
[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 0852.76066 · Zbl 0852.76066 · doi:10.1090/S0025-5718-96-00665-5
[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 0967.76064 · Zbl 0967.76064 · doi:10.1002/(SICI)1097-0363(20000515)33:1<1::AID-FLD978>3.0.CO;2-0
[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 0672.76018 · Zbl 0672.76018 · doi:10.1016/0045-7930(88)90024-2
[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 0528.76044 · Zbl 0528.76044 · doi:10.2514/3.8284
[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 0706.76075 · Zbl 0706.76075 · doi:10.1016/0045-7825(90)90095-4
[23] R. Verfürth , Error estimates for some quasi-interpolation operators . ESAIM: M2AN 33 ( 1999 ) 695 - 713 . Numdam | Zbl 0938.65125 · Zbl 0938.65125 · doi:10.1051/m2an:1999158
[24] R. Verfürth , A note on polynomial approximation in Sobolev spaces . ESAIM: M2AN 33 ( 1999 ) 715 - 719 . Numdam | Zbl 0936.41006 · Zbl 0936.41006 · doi:10.1051/m2an:1999159
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