Convergence of the MAC Scheme for the Compressible Stokes Equations

We prove in this paper the convergence of the Marker-and-Cell (MAC) scheme for the discretization of the steady state compressible Stokes equations on two- or three-dimensional Cartesian grids. The existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.