Inf-Sup Stability of the Discrete Duality Finite Volume method for the 2D Stokes problem

Franck Boyer (1) , Stella Krell (2, 3) , Flore Nabet (1)

1 I2M - Institut de Mathématiques de Marseille
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 COFFEE - COmplex Flows For Energy and Environment

''Discrete Duality Finite Volume'' schemes (DDFV for short) on general 2D meshes, in particular non conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretical point of view, whether or not the stability condition holds in this framework for various kind of mesh families. We obtain that different behaviors may occur depending on the geometry of the meshes. For instance, for conforming acute triangle meshes, we prove the unconditional Inf-Sup stability of the scheme, whereas for some conforming or non-conforming Cartesian meshes we prove that Inf-Sup stability holds up to a single unstable pressure mode. In any cases, the DDFV method appears to be very robust.

Mots clés: Inf-Sup stability.; DDFV methods; Stokes problem; Inf-Sup stability; Finite-volume methods

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Franck Boyer, Stella Krell, Flore Nabet. Inf-Sup Stability of the Discrete Duality Finite Volume method for the 2D Stokes problem. Mathematics of Computation, 2015, 84, pp.2705--2742. ⟨10.1090/mcom/2956⟩. ⟨hal-00795362v2⟩