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Our goal is to predict the band structure of photonic crystals. This taskrequires us to compute a number of the smallest non-zero eigenvalues of thetime-harmonic Maxwell operator depending on the chosen Bloch boundaryconditions.  We propose to use a block inverse iteration preconditioned with a suitablymodified geometric multigrid method. Since we are only interested in non-zeroeigenvalues, we eliminate the large null space by combining a lifting operatorand a secondary multigrid method. To obtain suitable initial guesses for theiteration, we employ a generalized extrapolation technique based on theminimization of the Rayleigh quotient that significantly reduces the number ofiteration steps and allows us to treat families of very large eigenvalueproblems efficiently.

Computing Maxwell eigenmodes with Bloch boundary conditions