Abstract
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.
Keywords: Incompressible Stokes equations, Finite volumes, Stability
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Eymard, R., Herbin, R., Latché, J. et al. On the stability of colocated clustered finite volume simplicial discretizations for the 2D Stokes problem. Calcolo 44, 219–234 (2007). https://doi.org/10.1007/s10092-007-0138-8
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DOI: https://doi.org/10.1007/s10092-007-0138-8