A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast

Abstract: In this paper, we suggest a new Heterogeneous Multiscale Method (HMM) for the Helmholtz equation with high contrast. The method is constructed for a setting as in Bouchitt\'e and Felbacq (C.R. Math. Acad. Sci. Paris 339(5):377--382, 2004), where the high contrast in the parameter leads to unusual effective parameters in the homogenized equation. We revisit existing homogenization approaches for this special setting and analyze the stability of the two-scale solution with respect to the wavenumber and the data. This includes a new stability result for solutions to the Helmholtz equation with discontinuous diffusion matrix. The HMM is defined as direct discretization of the two-scale limit equation. With this approach we are able to show quasi-optimality and an a priori error estimate under a resolution condition that inherits its dependence on the wavenumber from the stability constant for the analytical problem. Numerical experiments confirm our theoretical convergence results and examine the resolution condition. Moreover, the numerical simulation gives a good insight and explanation of the physical phenomenon of frequency band gaps.