  Wave propagation problems are notoriously difficult to solve. Time-harmonic
problems are especially challenging in mid and high frequency regimes. The main
reason is the oscillatory nature of solutions, meaning that the number of
degrees of freedom after discretisation increases drastically with the wave
number, giving rise to large complex-valued problems to solve. Additional
difficulties occur when the problem is defined in a highly heterogeneous
medium, as is often the case in realistic physical applications. For
time-discretised problems of Maxwell type, the main challenge remains the
significant kernel in curl-conforming spaces, an issue that impacts on the
design of robust preconditioners. This has already been addressed theoretically
for a homogeneous medium but not yet in the presence of heterogeneities. In
this review we provide a big-picture view of the main difficulties encountered
when solving wave propagation problems, from the first step of their
discretisation through to their parallel solution using two-level methods, by
showing their limitations on a few realistic examples. We also propose a new
preconditioner inspired by the idea of subspace decomposition, but based on
spectral coarse spaces, for curl-conforming discretisations of Maxwell's
equations in heterogeneous media.
