Stability analysis of heterogeneous Helmholtz problems and finite element solution based on propagation media approximation

Abstract

The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscilla-tory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.