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

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.

References in corpus (4)

• Kinetic description of diffusion-limited reactions in random catalytic media
• Theory of diffusion-influenced reactions in complex geometries
• The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains
• Reaction rate of a composite core-shell nanoreactor with multiple, spatially distributed embedded nano-catalysts

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