论文标题
通过非局部弛豫的非差异形式的线性椭圆方程的单调网格无网际方法
Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation
论文作者
论文摘要
我们通过非局部弛豫方法在点云上设计了一种无差异形式的线性椭圆方程的单调无线有限差方法。关键思想是PDE问题的非局部积分松弛与点云上强大的网格离散化的新颖组合。通过局部$ L_1 $ -Type优化过程获得最小的正模板，该过程自动保证了稳定性，因此，线性椭圆方程的网格离散化收敛性。一个主要的理论贡献是给定点云几何形状的一致和正模板存在。我们通过在每个内部点附近的椭圆形（2D）或椭圆形（2D）或椭圆形（3D）中找到邻居来提供足够的条件，从而通过Seibold概括了Poisson方程的研究（Comput Mechs Methods Appl Mech Eng 198（3-4）：592-601：592-601，2008）。众所周知，通常需要宽模型来构建线性椭圆方程的一致和单调有限差方案。我们的结果代表了与以前在该区域中以前已知的作品相比，在近定位状态下（当椭圆度常数变小时，椭圆度变为较小时）的正线性椭圆方程的正类差异方法的模块宽度估计值有显着改善。提供数值算法和实用指导，以关注椭圆度常数较小的情况。最后，我们为2D和3D的方法提供了数值结果，并检查了包括近定位的一系列椭圆形常数。
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.
