  This paper rigorously analyses preconditioners for the time-harmonic Maxwell
equations with absorption, where the PDE is discretised using curl-conforming
finite-element methods of fixed, arbitrary order and the preconditioner is
constructed using Additive Schwarz domain decomposition methods. The theory
developed here shows that if the absorption is large enough, and if the
subdomain and coarse mesh diameters and overlap are chosen appropriately, then
the classical two-level overlapping Additive Schwarz preconditioner (with PEC
boundary conditions on the subdomains) performs optimally -- in the sense that
GMRES converges in a wavenumber-independent number of iterations -- for the
problem with absorption. An important feature of the theory is that it allows
the coarse space to be built from low-order elements even if the PDE is
discretised using high-order elements. It also shows that additive methods with
minimal overlap can be robust. Numerical experiments are given that illustrate
the theory and its dependence on various parameters. These experiments motivate
some extensions of the preconditioners which have better robustness for
problems with less absorption, including the propagative case. At the end of
the paper we illustrate the performance of these on two substantial
applications; the first (a problem with absorption arising from medical
imaging) shows the empirical robustness of the preconditioner against
heterogeneity, and the second (scattering by a COBRA cavity) shows good
scalability of the preconditioner with up to 3,000 processors.
