  In this paper we give new results on domain decomposition preconditioners for
GMRES when computing piecewise-linear finite-element approximations of the
Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with
absorption parameter $\varepsilon \in \mathbb{R}$. Multigrid approximations of
this equation with $\varepsilon \not= 0$ are commonly used as preconditioners
for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for
such (so-called "shifted Laplace") preconditioners, either for the pure
Helmholtz equation, or even the absorptive equation ($\varepsilon \not=0$), is
still missing. We present a new theory for the absorptive equation that
provides rates of convergence for (left- or right-) preconditioned GMRES, via
estimates of the norm and field of values of the preconditioned matrix. This
theory uses a $k$- and $\varepsilon$-explicit coercivity result for the
underlying sesquilinear form and shows, for example, that if $|\varepsilon|\sim
k^2$, then classical overlapping additive Schwarz will perform optimally for
the absorptive problem, provided the subdomain and coarse mesh diameters are
carefully chosen. Extensive numerical experiments are given that support the
theoretical results. The theory for the absorptive case gives insight into how
its domain decomposition approximations perform as preconditioners for the pure
Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a
(scalable) multilevel preconditioner for the pure Helmholtz problem that has an
empirical computation time complexity of about $\mathcal{O}(n^{4/3})$ for
solving finite element systems of size $n=\mathcal{O}(k^3)$, where we have
chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect.
Experiments on problems with $h\sim k^{-1}$, i.e. a fixed number of grid points
per wavelength, are also given.
