  This work presents qualitative and numerical results on a system of partial
differential equations (PDEs) which models certain fluid-fluid interaction
dynamics. This system models a compressible fluid in a domain $\Omega^+ \subset
\mathbb{R}^2$, coupled to an incompressible fluid modeled by Stokes flow in
domain $\Omega^- \subset \mathbb{R}^2$, with the strong coupling implemented
through certain boundary conditions on the shared interface, $\Gamma$. The
wellposedness of this system is established by means of constructing for it a
semigroup generator representation. This representation is accomplished by
eliminating one of the pressure variables via identifying it as the solution of
a certain boundary value problem, while the wellposedness is established via a
nonstandard usage of the Babuska-Brezzi Theorem. In later sections, we
demonstrate how the earlier constructive proof of wellposedness naturally lends
itself to a certain finite element method (FEM), by which to numerically
approximate solutions of the given coupled PDE system. This FEM is provided
with error estimates and associated rates of convergence.
