  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.
