In the method of fundamental solutions (MFS), source nodes on circles outside the solution domains $S$ has been widely applied in numerical computation. In this paper, source nodes on an ellipse are proposed for solving Laplace’s equation using the MFS, and a robust error and stability analysis is established. Bounds on errors and condition numbers are derived for bounded simply-connected domains. Polynomial convergence rates can be achieved, but the exponential growth of the condition number is also obtained. In previous stability analysis of the MFS, circulant matrices have been always employed. This is the first time the stability analysis is explored for non-circular pseudo-boundaries, based on new techniques without using circulant matrices. The criteria for evaluating numerical techniques are provided, and some strategies for choosing pseudo-boundaries are suggested.

在基本解决方案（MFS）方法中，解决方案域之外的圆上的源节点 $\mathrm{小号}$已广泛应用于数值计算中。本文提出了一个椭圆上的源节点，用MFS求解拉普拉斯方程，并建立了鲁棒的误差和稳定性分析。对于有界的简单连接域，得出错误和条件编号的界线。可以实现多项式收敛速度，但是条件数的指数增长也可以得到。在先前的MFS稳定性分析中，始终使用循环矩阵。这是基于新技术而不使用循环矩阵的非圆形伪边界稳定性分析的首次探索。提供了评估数值技术的标准，并提出了一些选择伪边界的策略。