Convergence of a mixed method for a semi-stationary compressible Stokes system

We propose and analyze a finite element method for a semi– stationary Stokes system modeling compressible fluid flow subject to a Navier– slip boundary condition. The velocity (momentum) equation is approximated by a mixed finite element method using the lowest order Nédélec spaces of the first kind. The continuity equation is approximated by a standard piecewise constant upwind discontinuous Galerkin scheme. Our main result states that the numerical method converges to a weak solution. The convergence proof consists of two main steps: (i) To establish strong spatial compactness of the velocity field, which is intricate since the element spaces are only div or curl conforming. (ii) To prove that the discontinuous Galerkin approximations converge strongly, which is required in view of the nonlinear pressure function. Tools involved in the analysis include a higher integrability estimate for the discontinuous Galerkin approximations, a discrete equation for the effective viscous flux, and various renormalized formulations of the discontinuous Galerkin scheme.