We study time-harmonic scattering by a periodic array of penetrable, high-contrast obstacles with small period, confined to a bounded Lipschitz domain. The strong contrast between the obstacles and the background induces subwavelength resonances. We derive a frequency-dependent effective model in the vanishing-period limit and prove quantitative convergence of the heterogeneous scattered wave to the effective scattered wave. We also identify the limiting set of scattering resonances and establish convergence rates. Finally, we establish convergence rates for the far-field pattern of the heterogeneous problem to that of the effective model.
Du et al. (Mon,) studied this question.