Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

Abstract

Consider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain S$$ S $$ . In the standard MFS, the source nodes are located on a closed contour outside the domain boundary Γ(=∂S)$$ \Gamma \left(=\partial S\right) $$ , which is called pseudo‐boundary. For circular, elliptic, and general closed pseudo‐boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial‐lines outside Γ$$ \Gamma $$ . Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde‐wise matrices on nodes xi∈[a,b]$$ {x}_i\in \left[a,b\right] $$ with 0