A shape-parameterized RBF-partition of unity technique for PDEs

Abstract

In this paper, we study a direct discretization technique based on a radial basis function partition of unity (RBF-PU) method, which is built to numerically solve partial differential equations (PDEs). Unlike commonly used shape parameter free polyharmonic spline kernels, in this work we focus on local radial kernels depending on the shape parameter associated with the basis functions. The resulting scheme generally leads to more flexibility and accuracy, in particular when a polynomial term is added to the local RBF expansion. To emphasize the benefits deriving from use of the direct approach, we also compare it with the RBF finite difference (RBF-FD) method both in terms of computational efficiency and accuracy. Numerical results show the method performance by solving some elliptic PDE problems.