  Many problems in electrical engineering or fluid mechanics can be modeled by
parabolic-elliptic interface problems, where the domain for the exterior
elliptic problem might be unbounded. A possibility to solve this class of
problems numerically is the non-symmetric coupling of finite elements (FEM) and
boundary elements (BEM) analyzed in [Egger, Erath, Schorr, arXiv:1711.08487,
2017]. If, for example, the interior problem represents a fluid, this method is
not appropriate since FEM in general lacks conservation of numerical fluxes and
in case of convection dominance also stability.
  A possible remedy to guarantee both is the use of the vertex-centered finite
volume method (FVM) with an upwind stabilization option. Thus we propose a
(non-symmetric) coupling of FVM and BEM for a semi-discretization of the
underlying problem. For the subsequent time discretization we introduce two
options: a variant of the backward Euler method which allows us to develop an
analysis under minimal regularity assumptions and the classical backward Euler
method. We analyze both, the semi-discrete and the fully discrete system, in
terms of convergence and error estimates. Some numerical examples illustrate
the theoretical findings and give some ideas for practical applications.
