Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

TL;DR

This paper develops a semi-analytical method using the generalized method of separation of variables to compute Green functions for Laplace problems in 3D domains with multiple spherical boundaries, enabling efficient solutions for various physical applications.

Contribution

It introduces a semi-analytical approach employing solid harmonics and addition theorems to efficiently compute Green functions in complex spherical domains with multiple boundaries.

Findings

The method provides explicit Green functions for various boundary conditions.

It demonstrates high efficiency for exterior problems with multiple spherical obstacles.

Applications include calculating reaction rates, escape probabilities, and other diffusion characteristics.

Abstract

We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.