Exponentially convergent multiscale methods for 2D high frequency heterogeneous Helmholtz equations

In this paper, we present a multiscale framework for solving the 2D Helmholtz equation in heterogeneous media without scale separation and in the high frequency regime where the wavenumber $k$ can be large. The main innovation is that our methods achieve a nearly exponential rate of convergence with respect to the computational degrees of freedom, using a coarse grid of mesh size $O(1/k)$ without suffering from the well-known pollution effect. The key idea is a non-overlapped domain decomposition and its associated coarse-fine scale decomposition of the solution space that adapts to the media property and wavenumber; this decomposition is inspired by the multiscale finite element method (MsFEM). We show that the coarse part is of low complexity in the sense that it can be approximated with a nearly exponential rate of convergence via local basis functions, due to the compactness of a restriction operator that maps Helmholtz-harmonic functions to their interpolation residues on edges, while the fine part is local such that it can be computed efficiently using the local information of the right hand side. The combination of the two parts yields the overall nearly exponential rate of convergence of our multiscale method. Our method draws many connections to multiscale methods in the literature, which we will comment in detail. We demonstrate the effectiveness of our methods theoretically and numerically; an exponential rate of convergence is consistently observed and confirmed. In addition, we observe the robustness of our methods regarding the high contrast in the media numerically.