  The construction of weak solutions to compressible Navier-Stokes equations
via a numerical method (including a rigorous proof of the convergence) is in a
short supply, and so far, available only for one sole numerical scheme
suggested in Karper [{\em Numer. Math.}, 125(3) : 441--510, 2013] for the no
slip boundary conditions and the isentropic pressure with adiabatic coefficient
$\gamma>3$. Here we consider the same problem for the general non zero
inflow-outflow boundary conditions, which is definitely more appropriate
setting from the point of view of applications, but which is essentially more
involved as far as the existence of weak solutions is concerned. There is a few
recent proofs of existence of weak solutions in this setting, but none of them
is performed via a numerical method. The goal of this paper is to fill this
gap.
  The existence of weak solutions on the continuous level requires several
tools of functional and harmonic analysis and differential geometry whose
numerical counterparts are not known. Our main strategy therefore consists in
rewriting of the numerical scheme in its variational form modulo remainders and
to apply and/or to adapt to the new variational formulation the tools developed
in the theoretical analysis. In addition to the result, which is new, the
synergy between numerical and theoretical analysis is the main originality of
the present paper.
