On a stabilized colocated finite volume scheme for the Stokes problem

Eymard, Robert; Herbin, Raphaèle; Latché, Jean Claude

ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 3, pp. 501-527.

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem. It has been developed following two main ideas. On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes. On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique. We prove that, under usual regularity assumptions for the solution (each component of the velocity in ${\mathrm{H}}^2\left(\Omega \right)$ and pressure in ${\mathrm{H}}^1\left(\Omega \right)$), the scheme is first order convergent in the usual finite volume discrete ${\mathrm{H}}^1$ norm and the ${\mathrm{L}}^2$ norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order. With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions. Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete ${\mathrm{L}}^2$ norm.

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