On the stability of colocated clustered finite volume simplicial discretizations for the 2D Stokes problem

Titre de la revue : Calcolo Volume : 44 N° : 4 Pagination : 219-234 Date de publication : 01/12/2007

In this paper, we give a new (and simpler) stability proof for a cell-centered colocated finite volume scheme for the 2D Stokes problem, which may be seen as a particular case of a wider class of methods analyzed in [10]. The definition of this scheme involves two grids. The coarsest is a triangulation of the computational domain by acute-angled simplices, called clusters. The control volumes grid is finer, built by cutting each cluster along the lines joining the mid-edge points to obtain four sub-triangles. By building a Fortin projection operator explicitly, we prove 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 a second step, we prove that a stabilization which involves pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual colocated discretization (i.e., with the cell-centered approximation for the velocity and the pressure). Lastly we give an interpretation of this stabilization as a -minimal stabilization procedure-, as introduced by Brezzi and Fortin.