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In this paper, we propose a discretization for the (nonlinearized)compressible Stokes problem with a linear equation of state $\rho=p$, based onCrouzeix-Raviart elements. The approximation of the momentum balance isobtained by usual finite element techniques. Since the pressure is piecewiseconstant, the discrete mass balance takes the form of a finite volume scheme,in which we introduce an upwinding of the density, together with two additionalstabilization terms. We prove {\em a priori} estimates for the discretesolution, which yields its existence by a topological degree argument, and thenthe convergence of the scheme to a solution of the continuous problem.

A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case