  We introduce dissipative solutions to the compressible Navier-Stokes system
with potential temperature transport motivated by the concept of Young
measures. We prove their global-in-time existence by means of convergence
analysis of a mixed finite element-finite volume method. If a classical
solution to the compressible Navier-Stokes system with potential temperature
transport exists, we prove the strong convergence of numerical solutions. Our
results hold for the full range of adiabatic indices including the physically
relevant cases in which the existence of global-in-time weak solutions is open.
