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Issue ESAIM: M2AN
Volume 43, Number 5, September-October 2009
Page(s) 889 - 927
DOI 10.1051/m2an/2009031
Published online 01 August 2009

ESAIM: M2AN 43 (2009) 889-927
DOI: 10.1051/m2an/2009031

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

Robert Eymard1, Raphaèle Herbin2, Jean-Claude Latché3 and Bruno Piar3

1  Université de Marne-la-Vallée, France. robert.eymard@univ-mlv.fr
2  Université de Provence, France. herbin@cmi.univ-mrs.fr
3  Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France. jean-claude.latche@irsn.fr; bruno.piar@irsn.fr

Received October 30, 2006. Published online August 1st, 2009.

Abstract
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 L2 norm.


Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D05, 76D07, 76M25

Key words: Finite volumes, collocated discretizations, Stokes problem, Navier-Stokes equations, incompressible flows, analysis


© EDP Sciences, SMAI 2009


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