Fast integral methods for the Neumann Green's function: applications to capture and signaling problems in two dimensions

TL;DR

This paper introduces a high-order, fast numerical method for computing Neumann Green's functions in two dimensions, enabling accurate solutions for complex geometries and applications in particle capture problems.

Contribution

The paper develops a novel high-order integral method that efficiently computes interior and exterior Neumann Green's functions for general planar domains, including singularity handling and integral constraints.

Findings

Method achieves high accuracy on simple geometries like disks and ellipses.

Demonstrates effectiveness in optimizing trap configurations for Brownian particle capture.

Provides a fast, high-order discretization approach for complex domain Green's functions.

Abstract

We present a high order numerical method for the solution of the Neumann Green's function in two dimensions. For a general closed planar curve, our computational method resolves both the interior and exterior Green's functions with the source placed either in the bulk or on the surface -- yielding four distinct functions. Our method exactly represents the singular nature of the Green's function by decomposing the singular and regular components. In the case of the interior function, we exactly prescribe an integral constraint which is necessary to obtain a unique solution given the arbitrary constant solution associated with Neumann boundary conditions. Our implementation is based on a fast integral method for the regular part of the Green's function which allows for a rapid and high order discretization for general domains. We demonstrate the accuracy of our method for simple geometries such as disks and ellipses where closed form solutions are available. To exhibit the usefulness of these new routines, we demonstrate several applications to open problems in the capture of Brownian particles, specifically, how the small traps or boundary windows should be configured to maximize the capture rate of Brownian particles.