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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.
Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be non-robust in the presence of Neumann boundary conditions. In this paper we overcome this issue by formulating the RBF-generated finite difference
Consider using the right-preconditioned generalized minimal residual (AB-GMRES) method, which is an efficient method for solving underdetermined least squares problems. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-condition
In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using
We present a novel greedy Gauss-Seidel method for solving large linear least squares problem. This method improves the greedy randomized coordinate descent (GRCD) method proposed recently by Bai and Wu [Bai ZZ, and Wu WT. On greedy randomized coordin
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $Omega$. One of the requirements leading to approximation robustness is to place the interpola