  We consider one-level additive Schwarz preconditioners for a family of
Helmholtz problems with absorption and increasing wavenumber $k$. These
problems are discretized using the Galerkin method with nodal conforming finite
elements of any (fixed) order on meshes with diameter $h = h(k)$, chosen to
maintain accuracy as $k$ increases. The action of the preconditioner requires
solution of independent (parallel) subproblems (with impedance boundary
conditions) on overlapping subdomains of diameter $H$ and overlap $\delta\leq
H$. The solutions of these subproblems are linked together using
prolongation/restriction operators defined using a partition of unity. In
numerical experiments (with $\delta \sim H$) for a model interior impedance
problem, we observe robust (i.e. $k-$independent) GMRES convergence as $k$
increases. This provides a highly-parallel, $k-$robust one-level domain
decomposition method. We provide supporting theory by studying the
preconditioner applied to a range of absorptive problems, $k^2\mapsto k^2+
\mathrm{i} \varepsilon$, with absorption parameter $\varepsilon$. Working in
the Helmholtz ``energy'' inner product, and using the underlying theory of
Helmholtz boundary-value problems, we prove a $k-$independent upper bound on
the norm of the preconditioned matrix, valid for all $\vert \varepsilon\vert
\lesssim k^2$. We also prove a strictly-positive lower bound on the distance of
the field of values of the preconditioned matrix from the origin which holds
when $\varepsilon/k$ is constant or growing arbitrarily slowly with $k$. These
results imply robustness of the preconditioner for the corresponding absorptive
problem as k increases and give theoretical support for the observed robustness
of the preconditioner for the pure Helmholtz problem.
