  This paper presents a new numerical method for the compressible Navier-Stokes
equations governing the flow of an ideal isentropic gas. To approximate the
continuity equation, the method utilizes a discontinuous Galerkin
discretization on piecewise constants and a basic upwind flux. For the momentum
equation, the method is a new combined discontinuous Galerkin and finite
element method approximating the velocity in the Crouzeix-Raviart finite
element space. While the diffusion operator is discretized in a standard
fashion, the convection and time-derivative are discretized using discontinuous
Galerkin on the element average velocity and a Lax-Friedrich type flux. Our
main result is convergence of the method to a global weak solution as
discretization parameters go to zero. The convergence analysis constitutes a
numerical version of the existence analysis of Lions and Feireisl.
