Numerical approximation of the general compressible Stokes problem

In this paper, we propose a discretization for the compressible Stokes problem with an equation of state of the form p = ϕ(ρ) (where p stands for the pressure, ρ for the density and ϕ is a superlinear nondecreasing function from R to R). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density, and two additional terms. We prove the existence of a discrete solution and the convergence of this approximate solution to a solution of the continuous problem.