A regularized method of fundamental solutions is presented. The method can handle Neumann and mixed boundary conditions as well without using a desingularization technique. Instead, the approach combines the regularized method of fundamental solutions with traditional finite difference techniques based on some inner collocation points located in the vicinity of the boundary collocation points. Nevertheless, the resulting method remains meshless. The method avoids the problem of singularity and can be embedded in a natural multi-level context. The method is illustrated via a numerical example.