  We propose a mixed finite element method for the motion of a strongly
viscous, ideal, and isentropic gas. At the boundary we impose a Navier-slip
condition such that the velocity equation can be posed in mixed form with the
vorticity as an auxiliary variable. In this formulation we design a finite
element method, where the velocity and vorticity is approximated with the div-
and curl- conforming Nedelec elements, respectively, of the first order and
first kind. The mixed scheme is coupled to a standard piecewise constant upwind
discontinuous Galerkin discretization of the continuity equation. For the time
discretization, implicit Euler time stepping is used. Our main result is that
the numerical solution converges to a weak solution as the discretization
parameters go to zero. The convergence analysis is inspired by the continuous
analysis of Feireisl and Lions for the compressible Navier-Stokes equations.
Tools used in the analysis include an equation for the effective viscous flux
and various renormalizations of the density scheme.
