The construction of efficient iterative solvers for the time-harmonic Maxwell equations at high-frequency is a challenging problem. Some of the difficulties that arise are similar to those encountered in the case of the Helmholtz equation. Here we investigate how two-level domain decomposition preconditioners recently proposed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view. We develop a new theory for the time-harmonic Maxwell equations with absorption, which physically corresponds to the case of dissipative materials with non zero conductivity. This theory provides rates of convergence for GMRES with a two-level overlapping Additive Schwarz (AS) preconditioner, explicit in the wavenumber, the absorption, the coarse-grid diameter, the subdomain diameter and the overlap size. In particular, if the absorption is large enough, and if the subdomain and coarse mesh diameters are chosen appropriately, GMRES preconditioned with the two-level domain decomposition preconditioner converges in a wavenumber-independent number of iterations. Extensive large scale numerical experiments are carried out not only in the setting covered by the theory, but also for the time-harmonic Maxwell equations without absorption, and with more efficient two-level preconditioners, considering for instance impedance transmission conditions at interfaces between subdomains. The numerical results include an example arising from microwave medical imaging for the detection of brain strokes, which shows the robustness of the preconditioner against heterogeneity.
