Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows

Eymard, Robert; Herbin, Raphaèle; Latché, Jean-Claude; Piar, Bruno

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 889-927.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e., in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem. An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh. Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms. Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L${}^2$ norm.

Classification : 65N12, 65N15, 65N30, 76D05, 76D07, 76M25
Mots clés : finite volumes, collocated discretizations, Stokes problem, Navier-Stokes equations, incompressible flows, analysis

[1] F. Archambeau, N. Méchitoua and M. Sakiz, Code saturne: A finite volume code for turbulent flows. International Journal of Finite Volumes 1 (2004), http://www.latp.univ-mrs.fr/IJFV/. | MR

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

[3] F. Boyer and P. Fabrie, Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles, Mathématiques et Applications 52. Springer-Verlag (2006). | MR | Zbl

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

[5] E. Chénier, R. Eymard and O. Touazi, Numerical results using a colocated finite-volume scheme on unstructured grids for incompressible fluid flows. Numer. Heat Transf. Part B: Fundam. 49 (2006) 259-276.

[6] E. Chénier, R. Eymard, R. Herbin and O. Touazi, Collocated finite volume schemes for the simulation of natural convective flows on unstructured meshes. Int. J. Num. Methods Fluids 56 (2008) 2045-2068. | MR | Zbl

[7] Y. Coudière, T. Gallouët and R. Herbin, Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations. ESAIM: M2AN 35 (2001) 767-778. | Numdam | MR | Zbl

[8] K. Deimling, Nonlinear functional analysis. Springer-Verlag (1985). | MR | Zbl

[9] R. Eymard and T. Gallouët, H-convergence and numerical schemes for elliptic equations. SIAM J. Numer. Anal. 41 (2003) 539-562. | MR | Zbl

[10] R. Eymard and R. Herbin, A new colocated finite volume scheme for the incompressible Navier-Stokes equations on general non-matching grids. C. R. Acad. Sci., Sér. I Math. 344 (2007) 659-662. | MR | Zbl

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

[12] R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems. C. R. Acad. Sci., Sér. I Math. 339 (2004) 299-302. | MR | Zbl

[13] R. Eymard, R. Herbin and J.C. Latché, On a stabilized colocated finite volume scheme for the Stokes problem. ESAIM: M2AN 40 (2006) 501-528. | Numdam | MR | Zbl

[14] R. Eymard, T. Gallouët, R. Herbin and J.-C. Latché, Analysis tools for finite volume schemes. Acta Mathematica Universitatis Comenianae 76 (2007) 111-136. | Zbl

[15] 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. 45 (2007) 1-36. | MR | Zbl

[16] R. Eymard, R. Herbin, J.C. Latché and B. Piar, On the stability of colocated clustered finite volume simplicial discretizations for the 2D Stokes problem. Calcolo 44 (2007) 219-234. | MR | Zbl

[17] L.P. Franca and R. Stenberg, Error analysis of some Galerkin Least Squares methods for the elasticity equations. SIAM J. Numer. Anal. 28 (1991) 1680-1697. | MR | Zbl

[18] T. Gallouët, R. Herbin and M.H. Vignal, Error estimates for the approximate finite volume solution of convection diffusion equations with general boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1935-1972. | MR | Zbl

[19] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag (1986). | MR | Zbl

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

[21] L.E. Payne and H.F. Weinberger, An optimal Poincaré-inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286-292. | MR | Zbl

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

[23] R. Temam, Navier-Stokes Equations, Studies in mathematics and its applications. North-Holland (1977). | MR | Zbl

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