Efficient Band Structure Computation of 3D Photonic Crystals: a Parallel Multigrid Method with Higher Order Modified Finite Elements for the 3D Periodic Eigenvalue Problem in the Curl-curl Formulation

The band structure computation turns into solving a family of Maxwell eigenvalue problems on the periodicity domain. The discretization is done by the finite element method with special higher order H(curl)- and H1-conforming modified elements. The eigenvalue problem is solved by a preconditioned iterative eigenvalue solver with a projection onto the divergence-free vector fields. As a preconditioner we use the parallel multigrid method with a special Hiptmair smoother.