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

Published online by Cambridge University Press:  01 August 2009

Robert Eymard ,
Raphaèle Herbin ,
Jean-Claude Latché  and
Bruno Piar
Robert Eymard
Affiliation:
Université de Marne-la-Vallée, France. [email protected]
Raphaèle Herbin
Affiliation:
Université de Provence, France. [email protected]
Jean-Claude Latché
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France. [email protected]; [email protected]
Bruno Piar
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France. [email protected]; [email protected]
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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.

Keywords

Finite volumescollocated discretizationsStokes problemNavier-Stokes equationsincompressible flowsanalysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 889 - 927
Copyright
© EDP Sciences, SMAI, 2009

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