The RBF-FD method: developments and applications

Radial Basis Function (RBF) methods have become a truly meshless alternative for the interpolation of multidimensional scattered data and the solution of partial differencial equations (PDEs) on irregular domains. The dependence on the distance between centers makes RBF methods conceptually simple and easy to implement in anys dimension or shape of the domain. There are two different formulations for the solution of PDEs: the global RBF method and the local RBF method. In the global RBF formulation, the approximate solution is computed in the functional space spanned by a set of translated RBFs. The coordinates of the solution in this space are obtained by collocation. This formulation yields dense differentiation matrices which are spectrally convergent independently of the distribution of RBF centers. The principal drawback is that, as the overall number of centers increases, the condition number of the collocation matrices also increases, what restricts the applicability of the method in practical problems ....