  We analyse parallel overlapping Schwarz domain decomposition methods for the
Helmholtz equation, where the subdomain problems satisfy first-order absorbing
(impedance) transmission conditions, and exchange of information between
subdomains is achieved using a partition of unity. We provide a novel analysis
of this method at the PDE level (without discretization). First, we formulate
the method as a fixed point iteration, and show (in dimensions 1,2,3) that it
is well-defined in a tensor product of appropriate local function spaces, each
with $L^2$ impedance boundary data. Given this, we then obtain a bound on the
norm of the fixed point operator in terms of the local norms of certain
impedance-to-impedance maps arising from local interactions between subdomains.
These bounds provide conditions under which (some power of) the fixed point
operator is a contraction. In 2-d, for rectangular domains and strip-wise
domain decompositions (with each subdomain only overlapping its immediate
neighbours), we present two techniques for verifying the assumptions on the
impedance-to-impedance maps which ensure power contractivity of the fixed point
operator. The first is through semiclassical analysis, which gives rigorous
estimates valid as the frequency tends to infinity. These results verify the
required assumptions for sufficiently large overlap. For more realistic domain
decompositions, we directly compute the norms of the impedance-to-impedance
maps by solving certain canonical (local) eigenvalue problems. We give
numerical experiments that illustrate the theory. These also show that the
iterative method remains convergent and/or provides a good preconditioner in
cases not covered by the theory, including for general domain decompositions,
such as those obtained via automatic graph-partitioning software.
