Fast solution of boundary integral equations with the generalized Neumann kernel

TL;DR

This paper introduces a fast, efficient numerical method combining Nyström discretization, GMRES, and FMM to solve boundary integral equations with the generalized Neumann kernel, effective for complex multiply connected domains.

Contribution

It presents a novel fast computational approach that significantly reduces complexity for solving boundary integral equations with the generalized Neumann kernel.

Findings

Method achieves high accuracy for complex domains

Computational complexity is reduced to O((m+1)n log n)

Effective for domains with high connectivity and close boundaries

Abstract

A fast method for solving boundary integral equations with the generalized Neumann kernel and the adjoint generalized Neumann kernel is presented. The method is based on discretizing the integral equations by the Nystr\"om method with the trapezoidal rule to obtain $(m+1)n\times(m+1)n$ linear systems where $m+1$ is the multiplicity of the multiply connected domain and $n$ is the number of nodes in the discretization of each boundary component. The obtained linear systems are solved by the generalized minimal residual (GMRES) method. Each iteration of the GMRES method requires a matrix-vector product which can be computed using the Fast Multipole Method (FMM). The complexity of the presented method is $O((m+1)n\ln n)$ for the integral equation with the generalized Neumann kernel and $O((m+1)n)$ for the integral equation with the adjoint generalized Neumann kernel. The presented numerical results illustrate that the presented method gives accurate results even for domains with high connectivity, domains with piecewise smooth boundaries, and domains with close boundaries.