Laurent series based RBF-FD method to avoid ill-conditioning

We propose a new approach to avoid the inherent ill-condition in the computation of RBF-FD weights, which is due to the fact that the RBF interpolation matrix is nearly singular. The new approach is based on the semi-analytical computation of the Laurent series of the inverse of the RBF interpolation matrix. Once the Laurent series is obtained, it can be used to compute the RBF-FD weights of any differential operator exactly without extra cost. The proposed method also provides analytical formulas for the RBF-FD weights in terms of the parameters involved in the problem. These formulas can be used to derive the exact dependence of the truncation error in the approximation of any differential operator of a given function. Furthermore, from the analysis presented here one can derive the values of the parameters involved in the problem for which the RBF interpolation matrix becomes ill-conditioned and, hence, for which the weights cannot be obtained numerically.