Overlap Splines and Meshless Finite Difference Methods

We consider overlap splines that are defined by connecting the patches of piecewise functions via common values at given finite sets of nodes, without using any partitions of the computational domain. It is shown that some classical finite difference methods can be interpreted as collocation with overlap splines. Moreover, several versions of the meshless finite difference methods, such as the RBF-FD method, are equivalent to the collocation or discrete least squares with overlap splines, for appropriately chosen patches.