The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

TL;DR

The paper introduces the D-RBF-PU method, a faster, simpler, and more efficient localized RBF approach for solving PDEs that avoids derivatives and utilizes discontinuous PU weights, showing promising numerical results.

Contribution

It presents a novel direct RBF-PU method that simplifies computations and improves efficiency by avoiding derivatives and employing discontinuous PU weights.

Findings

Faster and simpler than standard RBF-PU methods

More accurate and efficient in some cases than RBF-FD

Effective on irregular 2D and 3D domains

Abstract

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 is benefited 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 2D and 3D domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.