Fast multipole method for the Laplace equation in half plane with Robin boundary condition

TL;DR

This paper introduces a fast multipole method tailored for the 2D Laplace equation in a half-plane with Robin boundary conditions, utilizing a novel Fourier-based expansion for efficient computation.

Contribution

It develops a new FMM framework incorporating a Fourier transform-based reaction component expansion for Robin boundary conditions in half-plane Laplace problems.

Findings

Achieves exponential convergence similar to free-space FMM

Demonstrates $O(N)$ computational complexity

Validates the method with numerical examples

Abstract

In this paper, we present a fast multipole method (FMM) for solving the two-dimensional Laplace equation in a half-plane with Robin boundary conditions. The method is based on a novel expansion theory for the reaction component of the Green's function. By applying the Fourier transform, the reaction field component is obtained in a Sommerfeld-type integral form. We derive far-field approximations and corresponding shifting and translation operators from the Fourier integral representation. The FMM for the reaction component is then developed by using the new far-field approximations incorporated into the classic FMM framework in which the tree structure is constructed from the original and image charges. Combining this with the standard FMM for the free-space components, we develop a fast algorithm to compute the interaction of the half plane Laplace Green's function. We prove that the method exhibits exponential convergence, similar to the free-space FMM. Finally, numerical examples are presented to validate the theoretical results and demonstrate that the FMM achieves $O(N)$ computational complexity.