The method of fundamental solutions for annular shaped domains

This paper consists of two parts. In the first part, we combine the analysis of Bogomolny [A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM Journal on Numerical Analysis 22 (1985) 644-669], Comodi and Mathon [M.I. Comodi, R. Mathon, A boundary approximation method for fourth order problems, Mathematical Models and Methods in Applied Sciences 1 (1991) 437-445] and Li, et al. [Z.C. Li, R. Mathon, P. Sermer, Boundary methods for solving elliptic equations with singularities, SIAM Journal on Numerical Analysis 24 (1987) 487-498] for the method of fundamental solutions (MFS), and new error bounds are derived for the bounded simply-connected in a readable approach. A factor of O(n) is removed in the error bounds of Bogomolny (1985). In the second part, we extend the analysis for the annular domain Sa by the MFS. The other fundamental solutions are needed, whose source nodes may also be located uniformly on a circle inside the domain Sa, as in the above reference. Error bounds are derived in detail to display the polynomial convergence rates.