THE DIRECT RADIAL BASIS FUNCTION PARTITION OF UNITY (D-RBF-PU) METHOD FOR SOLVING PDEs

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the "direct RBF partition of unity (D-RBF-PU)" method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular two- and three-dimensional domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.