# A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs

## TL;DR

This paper introduces a least squares RBF-PUM approach for solving PDEs that improves stability, reduces sensitivity to node layout, and enhances computational efficiency compared to traditional collocation methods.

## Contribution

It proposes a novel least squares formulation for RBF-PUM that controls conditioning via oversampling and removes node layout sensitivity.

## Key findings

- The least squares RBF-PUM is 5-10 times faster than collocation for the same accuracy.
- Theoretical error estimates are derived and confirmed by numerical experiments.
- The method maintains high order convergence rates under refinement.

## Abstract

Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUM. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to be 5-10 times faster than the collocation formulation for the same accuracy.