2017/02/23 by Elisabeth Larsson, Larsson, Elisabeth, Victor Shcherbakov +4 · 2 citations
Engineering · Mathematics · #65N12 #65N35 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.1702.07148
openalex publication_date 2017/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, collocation based radial basis function (RBF) partition of unity\nmethods (PUM) for solving partial differential equations have been formulated\nand investigated numerically and theoretically. When combined with stable\nevaluation methods such as the RBF-QR method, high order convergence rates can\nbe achieved and sustained under refinement. However, some numerical issues\nremain. The method is sensitive to the node layout, and condition numbers\nincrease with the refinement level. Here, we propose a modified formulation\nbased on least squares approximation. We show that the sensitivity to node\nlayout is removed and that conditioning can be controlled through oversampling.\nWe derive theoretical error estimates both for the collocation and least\nsquares RBF-PUM. Numerical experiments are performed for the Poisson equation\nin two and three space dimensions for regular and irregular geometries. The\nconvergence experiments confirm the theoretical estimates, and the least\nsquares formulation is shown to be 5-10 times faster than the collocation\nformulation for the same accuracy.\n