Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation

TL;DR

This paper introduces a novel monotone meshfree finite difference method for linear elliptic equations in non-divergence form, combining nonlocal relaxation with optimized stencils to ensure stability and convergence on point clouds.

Contribution

It presents a new approach for constructing positive stencils using local $l_1$-optimization, improving stencil width estimates especially in near-degenerate regimes.

Findings

Method guarantees stability and convergence for linear elliptic equations.

Provides sufficient conditions for positive stencil existence based on local geometry.

Demonstrates effective numerical performance in 2D and 3D, including near-degenerate cases.

Abstract

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.