Issue |
ESAIM: M2AN
Volume 43, Number 5, September-October 2009
|
|
---|---|---|
Page(s) | 889 - 927 | |
DOI | https://doi.org/10.1051/m2an/2009031 | |
Published online | 01 August 2009 |
Convergence analysis of a locally stabilized collocated finite volume scheme for incompressible flows
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:
30
October
2006
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.