三维不连通球边界区域中Laplacian Green函数的半解析计算
Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries
摘要：
我们应用广义分离变量法（GMSV）求解三维区域中具有不连通球形边界的Laplace算子的边值问题（即，不重叠的部分反应球形汇或障碍物的任意配置）。我们考虑外部和内部问题以及所有最常见的边界条件：Dirichlet，Neumann，Robin和共轭边界条件。利用平移加法定理的固体谐波之间切换的局部球坐标，我们得到的半解析表达式的格林函数的部分解的线性组合，其系数是固定的边界条件。虽然数值计算的系数涉及系列截断和矩阵求逆，使用的固体谐波作为基础函数自然适应的内在对称性的问题，使GMSV特别有效，特别是外部问题。得到的格林函数是解决边值问题和确定稳态扩散的各种特性的关键成分，例如反应速率、逃逸概率、调和测度、停留时间和平均首次通过时间，仅举几例。数值实现的相关方面和潜在的应用在化学物理，传热，静电学和流体力学进行了讨论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
