# Fast Multipole Method For 3-D Helmholtz Equation In Layered Media

## TL;DR

This paper introduces a fast multipole method tailored for efficiently computing wave interactions in 3-D layered media, combining classic FMM with new expansions for reaction fields, achieving near-linear complexity.

## Contribution

It develops a novel FMM that integrates reaction field components for layered media using new multipole and local expansions, improving efficiency over traditional free space FMM.

## Key findings

- Achieves $O(N)$ complexity for low-frequency wave interactions.
- Demonstrates fast convergence of multipole expansions.
- Validates efficiency through numerical experiments.

## Abstract

In this paper, a fast multipole method (FMM) is proposed to compute long-range interactions of wave sources embedded in 3-D layered media. The layered media Green's function for the Helmholtz equation, which satisfies the transmission conditions at material interfaces, is decomposed into a free space component and four types of reaction field components arising from wave reflections and transmissions through the layered media. The proposed algorithm is a combination of the classic FMM for the free space component and FMMs specifically designed for the four types reaction components, made possible by new multipole expansions (MEs) and local expansions (LEs) as well as the multipole-to-local translation (M2L) operators for the reaction field components. { Moreover, equivalent polarization source can be defined for each reaction component based on the convergence analysis of its ME. The FMMs for the reaction components, implemented with the target particles and equivalent polarization sources, are found to be much more efficient than the classic FMM for the free space component due to the fact that the equivalent polarization sources and the target particles are always separated by a material interface.} As a result, the FMM algorithm developed for layered media has a similar computational cost as that for the free space. Numerical results validate the fast convergence of the MEs and the $O(N)$ complexity of the FMM for interactions of low-frequency wave sources in 3-D layered media.