A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I: The isothermal case

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Abstract
In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state ρ = p \rho =p , based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove a priori estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.

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