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

TL;DR

This work develops a fast, high-order boundary-integral method to compute the Neumann Green's function in two dimensions for general planar domains, simultaneously handling interior and exterior problems with bulk and surface sources. By decomposing the singular part and enforcing a zero-average interior constraint, the method yields four accurately evaluated Green's functions and enables precise gradients, Hessians, and optimization of trap configurations. The approach is validated against disk and ellipse benchmarks and used to optimize interior traps and boundary windows in complex geometries, with direct applications to Brownian capture and signal-sensing problems. The resulting framework provides a versatile tool for MFPT/GMFPT analysis, directional sensing, and trap optimization in realistic domains, with clear paths for extensions to 3D and time-dependent statistics.

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.

Figures (15)

• Figure 1: We consider Brownian motion both interior and exterior to a planar geometry $\Omega$. For the interior diffusion, a long standing problem is to find configurations of interior traps or locations of boundary windows that minimize the capture times. For the exterior problem, we seek the capture rate at boundary windows on $\partial\Omega$. To address these questions, we develop numerical solutions of the interior $G^{\textrm{int}}$ and exterior $G^{\textrm{ext}}$ Neumann Green's function for both surface sources (on $\partial\Omega$) and bulk (off $\partial\Omega$) sources (red dots). In total, we provide high accuracy methods for evaluating four Green's functions as seen in the above plots of $|G^{\textrm{int}}_{s,b}|$ and $|G^{\textrm{ext}}_{s,b}|$. \newlabelfig:intro0
• Figure 2: Convergence of the relative errors in the case of the unit disk against panel number and quadrature order $k$. Numerical approximation of the regular part $R^{\textrm{int}}_b$ (Panel (a)), the gradient $\|\nabla R^{\textrm{int}}_b\|_2$ (Panel (b)) and the Hessian $\|\nabla^2 R^{\textrm{int}}_b \|_2$ (Panel (c)) for the point $\emph{y} = (\frac{1}{4},\frac{1}{3})$. \newlabelfig:ConvergenceDisk0
• Figure 3: Errors for the surface Green's function for the disk. Panel (a) shows the absolute error in $R^{\textrm{int}}_s(\emph{x;y})$ with the exact solution given in \ref{['eq:GreensDiskSurf']}. Panel (b) shows the absolute error in $R^{\textrm{ext}}_s(\emph{x;y})$ with the exact solution given in \ref{['eq:GreensDiskSurfExt']}. In both cases the source point is $\emph{y} = [0,1]$ (solid red dot). \newlabelfig:Disk_surf0
• Figure 4: Convergence of relative errors of $R^{\textrm{int}}_b\emph{(y;y)}$ for elliptical domains $(x_1/a)^2 + (x_2/b)^2 \leq1$ and quadrature orders $k$. The point $\emph{y} =[a/4,b/3]$ with $|\Omega| = \pi a b = \pi$ and various values of $a$ ($b=1/a$). \newlabelfig:ellipseConvergence0
• Figure 5: Convergence of the regular parts of the interior $(R^{\textrm{int}}_{s}$ - upper row) and exterior $(R^{\textrm{ext}}_{s}$ - lower row) surface Green's functions for the elliptical domain $(x_1/a)^2 + (x_2/b)^2 = 1$. The area of the ellipses are fixed at $|\Omega| = \pi$ ($b = 1/a$) and from left to right results are presented for $a = 1.5$, $a= 2.0$ and $a= 2.5$. The method is able to reduce relative errors to roughly $\mathcal{O}(10^{-12})$. \newlabelfig:surfaceGreens0