Treffer: An $Hp$-Adaptive Sampling Algorithm for Dispersion Relation Reconstruction of 3d Photonic Crystals: An \textit{hp}-adaptive sampling algorithm for dispersion relation reconstruction of 3D photonic crystals

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.