Improved stencil selection for meshless finite difference methods in 3D

We introduce a geometric stencil selection algorithm for Laplacian in 3D that signifi-cantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding a given point that admits an accurate numerical differentiation formula. The subset serves as an influence set for the numerical approximation of the Laplacian in meshless finite difference methods using either polynomial or kernel-based techniques. Numerical experiments demonstrate a competitive performance of this method in comparison to the finite element method and to other selection methods for solving the Dirichlet problems for the Poisson equation on several STL models. Discretization nodes for these domains are obtained either by 3D triangulations or from Cartesian grids or Halton quasi-random sequences.(c) 2022 Elsevier B.V. All rights reserved.

Automated summary

We present a geometric stencil selection algorithm for the Laplacian in 3D, which improves upon octant-based selection methods. The algorithm aims to choose a small subset of irregular points surrounding a given point to obtain an accurate numerical differentiation formula. This subset is utilized as an influence set for numerical approximations of the Laplacian using polynomial or kernel-based techniques. Numerical experiments demonstrate the competitiveness of this method in solving the Dirichlet problems for the Poisson equation on various STL models. Discretization nodes are obtained through 3D triangulations, Cartesian grids, or Halton quasi-random sequences.