Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries

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.