We design a monotone meshfree finite difference method for linear elliptic equations in the non-divergence form on point clouds via a nonlocal relaxation method. The key idea is a novel combination of a nonlocal integral relaxation of the PDE problem with a robust meshfree discretization on point clouds. Minimal positive stencils are obtained through a local $l_1$-type optimization procedure that automatically guarantees the stability and, therefore, the convergence of the meshfree discretization for linear elliptic equations. A major theoretical contribution is the existence of consistent and positive stencils for a given point cloud geometry. We provide sufficient conditions for the existence of positive stencils by finding neighbors within an ellipse (2d) or ellipsoid (3d) surrounding each interior point, generalizing the study for Poisson's equation by Seibold (Comput Methods Appl Mech Eng 198(3-4):592-601, 2008). It is well-known that wide stencils are in general needed for constructing consistent and monotone finite difference schemes for linear elliptic equations. Our result represents a significant improvement in the stencil width estimate for positive-type finite difference methods for linear elliptic equations in the near-degenerate regime (when the ellipticity constant becomes small), compared to previously known works in this area. Numerical algorithms and practical guidance are provided with an eye on the case of small ellipticity constant. At the end, we present numerical results for the performance of our method in both 2d and 3d, examining a range of ellipticity constants including the near-degenerate regime.

翻译：本文通过非局部松弛方法，为点云上非散度形式的线性椭圆方程设计了一种单调无网格有限差分方法。核心思想是将PDE问题的非局部积分松弛与鲁棒的点云无网格离散化进行新颖结合。通过局部$l_1$型优化过程获取最小正模板，该优化过程自动保证线性椭圆方程无网格离散化的稳定性，进而确保收敛性。一项主要理论贡献是针对给定点云几何结构的一致正模板存在性。我们通过为每个内点周围的椭圆（二维）或椭球（三维）内寻找邻点，给出了正模板存在的充分条件，将Seibold（Comput Methods Appl Mech Eng 198(3-4):592-601, 2008）关于泊松方程的研究进行了推广。众所周知，构造线性椭圆方程的一致单调有限差分格式通常需要宽模板。我们的结果显著改善了近退化情形（当椭圆性常数较小时）下正型有限差分方法中模板宽度的估计，优于该领域此前已知的研究工作。本文提供了数值算法与实践指导，特别关注小椭圆性常数的情形。最后，我们给出了方法在二维和三维空间中的数值结果，检验了包括近退化情形在内的椭圆性常数范围。