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

The generalized method of separation of variables (GMSV) is applied to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (e.g., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We...

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Bibliographic Details
Published in:Journal of computational physics Vol. 379; pp. 91 - 117
Main Authors: Grebenkov, Denis S., Traytak, Sergey D.
Format: Journal Article
Language:English
Published: Cambridge Elsevier Inc 15.02.2019
Elsevier Science Ltd
Elsevier
Subjects:
Approximation
Basis functions
Biological Physics
Boundary conditions
Boundary value problem
Boundary value problems
Chemical Physics
Computational fluid dynamics
Computational Physics
Condensed Matter
Diffusion rate
Diffusion–reaction
Dirichlet problem
Domains
Electrostatics
Fluid flow
Green function
Green's functions
Harmonics
Heat sinks
Hydrodynamics
Laplace operator
Linear algebra
Mathematical Physics
Numerical analysis
Operators (mathematics)
Organic chemistry
Physics
Semi-analytical solution
Spherical coordinates
Statistical Mechanics
Laplace operator
Semi-analytical solution
Boundary value problem
Diffusion–reaction
Green function
diffusion-reaction
boundary value problem
semi-analytical solution
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:The generalized method of separation of variables (GMSV) is applied to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (e.g., 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 solution of a system of linear algebraic equations, 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. •Semi-analytical form of the Laplacian Green function is obtained.•Exterior and interior boundary value problems with Dirichlet, Neumann, Robin, and conjugate conditions are treated.•Mesh-free computation, very rapid convergence, no need for an outer boundary.•Various characteristics of stationary diffusion are deduced.•A Matlab package is released and freely available.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.10.033