  In this work we investigate the computation of dispersion relation (i.e.,
band functions) for three-dimensional photonic crystals, formulated as a
parameterized Maxwell eigenvalue problem, using a novel hp-adaptive sampling
algorithm. We develop an adaptive sampling algorithm in the parameter domain
such that local elements with singular points are refined at each iteration,
construct a conforming element-wise polynomial space on the adaptive mesh such
that the distribution of the local polynomial spaces reflects the regularity of
the band functions, and define an element-wise Lagrange interpolation operator
to approximate the band functions. We rigorously prove the convergence of the
algorithm. To illustrate the significant potential of the algorithm, we present
two numerical tests with band gap optimization.
