# Windowed Green Function method for the Helmholtz equation in presence of multiply layered media

## TL;DR

This paper introduces a fast, accurate, and flexible Windowed Green Function method for solving acoustic and electromagnetic scattering problems in layered media, avoiding expensive integral computations.

## Contribution

The paper develops a novel WGF approach that efficiently evaluates oscillatory integrals in layered media without Sommerfeld integrals, improving speed and accuracy.

## Key findings

- Numerical errors decrease faster than any negative power of window size.
- Method is up to thousands of times faster than traditional Sommerfeld integral methods.
- High accuracy achieved with less computational expense.

## Abstract

This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in presence an arbitrary number of penetrable layers. Relying on use of certain slow-rise windowing functions, the proposed Windowed Green Function approach (WGF) efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multi-layer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (SIAM J. Appl. Math. 76(5), p. 1871, 2016), is fast, accurate, flexible, and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on use of Sommerfeld integrals.