Exponential convergence for multipole and local expansions and their translations for sources in layered media: 2-D acoustic wave
This work provides a theoretical foundation for fast multipole methods in layered media, which is important for computational electromagnetics and acoustics.
The paper provides a derivation and rigorous proof of exponential convergence for multipole and local expansions and their translations in 2-D layered media for acoustic waves, showing convergence depends on a polarized distance.
In this paper, we will first give a derivation of the multipole expansion (ME) and local expansion (LE) for the far field from sources in general 2-D layered media and the multipole-to-local translation (M2L) operator by using the generating function for Bessel functions. Then, we present a rigorous proof of the exponential convergence of the ME, LE, and M2L for 2-D Helmholtz equations in layered media. It is shown that the convergence of ME, LE, and M2L for the reaction field component of the Green's function depends on a polarized distance between the target and a polarized image of the source.