  We consider one-level additive Schwarz domain decomposition preconditioners
for the Helmholtz equation with variable coefficients (modelling wave
propagation in heterogeneous media), subject to boundary conditions that
include wave scattering problems. Absorption is included as a parameter in the
problem. This problem is discretised using $H^1$-conforming nodal finite
elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen
so that the error remains bounded with increasing $k$. The action of the
one-level preconditioner consists of the parallel solution of problems on
subdomains (which can be of general geometry), each equipped with an impedance
boundary condition. We prove rigorous estimates on the norm and field of values
of the left- or right-preconditioned matrix that show explicitly how the
absorption, the heterogeneity in the coefficients and the dependence on the
degree enter the estimates. These estimates prove rigorously that, with enough
absorption and for $k$ large enough, GMRES is guaranteed to converge in a
number of iterations that is independent of $k,p,$ and the coefficients. The
theoretical threshold for $k$ to be large enough depends on $p$ and on the
local variation of coefficients in subdomains (and not globally). Extensive
numerical experiments are given for both the absorptive and the propagative
cases; in the latter case we investigate examples both when the coefficients
are nontrapping and when they are trapping. These experiments (i) support our
theory in terms of dependence on polynomial degree and the coefficients; (ii)
support the sharpness of our field of values estimates in terms of the level of
absorption required.
