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

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