Abstract For Laplace's equation in a bounded simply-connected domain Ω in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al. [10] , the theoretical analysis is much behind (Li [23] only for unit sphere Ω). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial convergence rates can be achieved. Numerical experiments are carried out to support the analysis made, and several useful locations of source nodes are investigated numerically. The analysis in this paper may lay a theoretical basis of the MFS for 3D problems, as Bogomolny [8] and [24] for 2D problems. Besides, the method of particular solutions (MPS) in [26] is also studied by using the spherical harmonic functions (SHF). The optimal polynomial convergence rates and the exponential growth of condition number (Cond) are obtained. The source nodes are located based on the abscissas of quadrature rules on surfaces; they are “grid-like”. Since most of 3D problems, in reality, can not be simplified to 2D problems, and since the MFS has more advantages for 3D problems in algorithm simplicity and wide application, the study in this paper is essential and important to the MFS.

中文翻译：

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用基解法和特解法求解三维拉普拉斯方程

摘要 本文针对三维有界单连通域Ω中的拉普拉斯方程，研究了基本解法(MFS)。虽然可以在 Chen 等人中找到一些数值计算。[10] ，理论分析落后很多（Li [23] 仅针对单位球体Ω）。我们致力于探索对 MFS 进行严格的错误分析。推导出误差界限，并且可以实现最优多项式收敛速度。进行了数值实验以支持所做的分析，并对源节点的几个有用位置进行了数值研究。本文的分析可以为 3D 问题的 MFS 奠定理论基础，如 Bogomolny [8] 和 [24] 的 2D 问题。此外，还使用球谐函数（SHF）研究了[26]中的特解法（MPS）。得到最优多项式收敛速度和条件数(Cond)的指数增长。源节点根据曲面上的正交规则的横坐标定位；它们是“网格状的”。由于大多数3D问题在现实中无法简化为2D问题，而MFS对于3D问题在算法简单性和广泛应用方面具有更多优势，因此本文的研究对MFS至关重要。