The Overlapped Radial Basis Function-Finite Difference (RBF-FD) Method for the Numerical Solution of PDEs.

We present a generalization of the RBF-FD method that allows full control of the overlap between RBF-FD stencils. We accomplish this by introducing a continuous overlap parameter $\delta \in [0,1]$ such that $\delta=1$ recovers the standard RBF-FD method and $\delta=0$ results in a full decoupling of the RBF-FD stencils. We show with a simple example that global interpolation with both RBFs and RBFs augmented with polynomials is superior to polynomial least squares in confining Runge oscillations to the boundary, thereby justifying retaining more RBF-FD weights from each RBF-FD stencil than just the center; this also provides greater insight into the success of the recent polyharmonic spline RBF-FD method with high-degree polynomials appended. We demonstrate the efficiency of the \emph{overlapped RBF-FD method} by solving parabolic differential equations in 2D and 3D. Our results show that the new method can achieve as high as a 20x speedup over the standard RBF-FD method in the task of forming differentiation matrices. Further, the overlapped RBF-FD method allows for stencil sizes up to 6x larger than those usable in the standard RBF-FD method for a lower total computational cost.