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 is removed in the error bounds of Bogomolny (1985). In the second part, we extend the analysis for the annular domain S_a by the MFS. The other fundamental solutions are needed, whose source nodes may also be located uniformly on a circle inside the domain S_a, as in the above reference. Error bounds are derived in detail to display the polynomial convergence rates.