  We present Nystr\"om discretizations of multitrace formulations and
non-overlapping Domain Decomposition Methods (DDM) for the solution of
Helmholtz transmission problems for bounded composite scatterers with piecewise
constant material properties. We investigate the performance of DDM with both
classical Robin and generalized Robin boundary conditions. The generalized
Robin boundary conditions incorporate square root Fourier multiplier
approximations of Dirichlet to Neumann operators. While the classical version
of DDM is not particularly well suited for Krylov subspace iterative solvers,
we show that the associated DDM linear system can be efficiently solved by
hierarchical elimination via Schur complements of the Robin data. We show
through numerical examples that the latter version of DDM gives rise to small
numbers of Krylov subspace iterations that depend mildly on the frequency and
number of subdomains.
