Consider Laplace’s equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index.

考虑有界单连通域S中的拉普拉斯方程，并使用基本解法 (MFS)。Dou等人对圆形/椭圆形伪边界进行了误差和稳定性分析。( J. Comp. Appl. Math . 377:112861, 2020)，得到条件数(Cond)的多项式收敛率和指数增长率。Dou 等人为更复杂的解域建议了通用伪边界。( J. Comp. Appl. Math.377:112861，2020，第 5 节）。由于病态严重，MFS 计算的成功主要取决于稳定性。本文致力于源节点平滑闭合伪边界的稳定性分析。导出了 Cond 的界限，并且还获得了指数增长率。本文首次探讨了非圆形/非椭圆形伪边界的 MFS 稳定性分析。循环矩阵常用于 MFS 的稳定性分析；但本文中的稳定性分析是基于新技术进行的，没有像 Dou 等人那样使用循环矩阵。( J. Comp. Appl. Math. 377:112861, 2020)。为了追求更好的伪边界，敏感性指数是从稳定性的增长/收敛率到准确性提出的。通过试验计算可以找到更好的 MFS 伪边界，以发展 Dou 等人的研究。( J. Comp. Appl. Math.377:112861, 2020) 用于选择伪边界。对于高度平滑和奇异的解，更好的伪边界是不同的；对敏感性指数的分析进行了探索。圆形/椭圆形伪边界对于高度平滑的解是最佳的，但不适用于奇异的解。在本文中，在计算中选择了类似变形虫的域。开发了几种有用的伪边界类型，它们的算法很简单，无需使用非线性解决方案。对于奇异解，通过灵敏度指数对不同的伪边界进行数值比较。