An hp-Adaptive Sampling Algorithm on Dispersion Relation Reconstruction for 2D Photonic Crystals

Computing the dispersion relation for two-dimensional photonic crystals is a notoriously challenging task: It involves solving parameterized Helmholtz eigenvalue problems with high-contrast coefficients. To resolve the challenge, we propose a novel hp-adaptive sampling scheme that can detect singular points via adaptive mesh refinement in the parameter domain, and meanwhile, allow for adaptively enriching the local polynomial spaces on the elements that do not contain singular points. In this way, we obtain an element-wise interpolation on an adaptive mesh. We derive an exponential convergence rate when the number of singular points is finite, and a first-order convergence rate otherwise. Numerical tests are provided to illustrate its performance.