  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 Nedelec 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.
