Computational high frequency scattering from high-contrast heterogeneous media

Abstract

This article considers the computational (acoustic) wave propagation in strongly heterogeneous structures beyond the assumption of periodicity. A high contrast between the constituents of microstructured multiphase materials can lead to unusual wave scattering and absorption, which are interesting and relevant from a physical viewpoint, for instance, in the case of crystals with defects. We present a computational multiscale method in the spirit of the Localized Orthogonal Decomposition and provide its rigorous a priori error analysis for two-phase diffusion coefficients that vary between 1 and very small values. Special attention is paid to the extreme regimes of high frequency, high contrast, and their previously unexplored coexistence. A series of numerical experiments confirms the theoretical results and demonstrates the ability of the multiscale approach to efficiently capture relevant physical phenomena.

A generalized finite element method for problems with sign-changing coefficients

“…However, the Helmholtz equation (with positive coefficients), was analyzed in [16,29,31]. In particular, it is shown that the corrector problems are well-posed under a resolution condition on because the 2 -perturbation in the Gårding inequality can be absorbed for functions in the kernel due to the property (3.5) of .…”

“…Proof. The well-posedness of (5.3) follows from an inf-sup condition on , (see [31] for instance). This directly yields quasi-optimality and the error estimate (5.4), where we refer to Chapter 2 of [24] for details.…”

“…i.e., quadratic convergence in the 2 (Ω)-norm. We refer to, e.g., [31] for details. Finally, we have that…”

“…The latter are defined as solutions of local fine-scale problems. Since its introduction in [20,25], the LOD has been successfully applied in various situations, where we mention in particular the reduction of the pollution effect for high-frequency Helmholtz problems [16,29,31]. The efficient implementation of the method is outlined in [15].…”

Multiscale scattering in nonlinear Kerr-type media

“…The main contribution of this article are the presentation and numerical analysis of multiscale methods in the spirit of the LOD for the Helmholtz equation with Kerr-type nonlinearity. Various works have successfully applied the LOD to wave propagation problems such as the wave equation [AH17,MP19], the Helmholtz equation with constant [GP15,Pet17] and spatially varying coefficients [BGP17,PV20] as well as time-harmonic Maxwell's equations [GHV18,Ver17,HP20]. Besides dealing with multiscale coefficients, the LOD can also reduce the well-known pollution effect for the linear Helmholtz equation [GP15,Pet17].…”