  We model time-harmonic acoustic scattering by an object composed of
piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose
and analyze new formulations that couple, adopting a Costabel-type approach,
boundary integral equations for the homogeneous subdomains with volume
variational formulations for the heterogeneous subdomain. This is an extension
of the Costabel FEM-BEM coupling to a multi-domain configuration, with
cross-points allowed, i.e. points where three or more subdomains are adjacent.
While generally just the exterior unbounded subdomain is treated with the BEM,
here we wish to exploit the advantages of BEM whenever it is applicable, that
is, for all the homogeneous parts of the scattering object. Our formulation is
based on the multi-trace formalism, which initially was introduced for acoustic
scattering by piece-wise homogeneous objects. Instead, here we allow the
wavenumber to vary arbitrarily in a part of the domain. We prove that the
bilinear form associated with the proposed formulation satisfies a G{\aa}rding
coercivity inequality, which ensures stability of the variational problem if it
is uniquely solvable. We identify conditions for injectivity and construct
modified versions immune to spurious resonances.
