  Our goal is to predict the band structure of photonic crystals. This task
requires us to compute a number of the smallest non-zero eigenvalues of the
time-harmonic Maxwell operator depending on the chosen Bloch boundary
conditions.
  We propose to use a block inverse iteration preconditioned with a suitably
modified geometric multigrid method. Since we are only interested in non-zero
eigenvalues, we eliminate the large null space by combining a lifting operator
and a secondary multigrid method. To obtain suitable initial guesses for the
iteration, we employ a generalized extrapolation technique based on the
minimization of the Rayleigh quotient that significantly reduces the number of
iteration steps and allows us to treat families of very large eigenvalue
problems efficiently.
