Fast Multipole Method For 3-D Helmholtz Equation in Layered Media

Abstract

In this paper, we propose a fast multipole method (FMM) for 3-D linearized Poisson-Boltzmann (PB) equation in layered media. The main framework of the algorithm is analogous to the FMM for Helmholtz and Laplace equation in layered media [1,2], using an extension of the Funk-Hecke formula for pure imaginary wave number. Moreover, a recurrence formula is provided for the run-time computation of the Sommerfeld-type integrals used in the FMM algorithm. Due to the similarity between Helmholtz and linearized PB equation, the recurrence formula can also be used for the FMM of Helmholtz equation in layered media with minor changes as mentioned in [1]. Numerical results validate that the FMM for interactions of charges under screen's potentials in layered media has the same accuracy and CPU complexity as the classic FMM for charge interactions in free space.

Exponential convergence for multipole and local expansions and their translations for sources in layered media: three-dimensional Laplace equation

“…First, we consider the direct MEs for reaction components. We prove that they have exponential convergence but with convergence rates depend on polarization distances as defined in [10] and then used by [9,11]. Therefore, the concept of equivalent polarization sources is crucial for the development of the FMMs for reaction components.…”

“…In applying the free space FMM framework to handle the reaction field components of the layered Green's function, the main problem is the MEs given in (3.22) are generally not compatible with the hierarchical tree structure design. In our previous work [9,10], we have introduced the conception of equivalent polarization sources to overcome this problem. The idea was inspired by our theoretical analysis for 2-dimensional Helmholtz equation (cf.…”

“…Recently, we have extended the FMMs of the Helmholtz, Laplace and Poisson-Botzmann equations from free space to layered media (cf. [10,12,9,11]). The new FMMs significantly enlarge the application area of the classic ones.…”

A Matrix Basis Formulation For The Green's Functions Of Maxwell's Equations And The Elastic Wave Equations In Layered Media

“…Remark 3.1. In the previous work of the Helmholtz equations [13,14,15], the separation of variable z ′ was also executed, so that each reaction component was further divided by the "propagating direction" of z ′ . Then, we re-format the restricting equations of the tensor potential G A , including the interface conditions and the radiation conditions, using the matrix basis…”

“…Third, the rotational symmetry allows us to apply fast solvers easily, e.g. the fast multipole method in layered media [13]. We also develop a vector basis formulation which is simplified from the matrix version, used for the LMDG of the mix-phase elastic wave equations where the source originates in fluid medium.…”