Minimal positive stencils in meshfree finite difference methods for the Poisson equation

TL;DR

This paper introduces a new meshfree finite difference method for the Poisson equation that produces minimal positive stencils, improving accuracy and efficiency over traditional least squares approaches.

Contribution

It develops a novel approach for generating minimal positive stencils in meshfree methods, with conditions ensuring their existence and improved computational performance.

Findings

Positive stencils are achievable with the new method.

The approach outperforms least squares in accuracy.

Computational efficiency is improved.

Abstract

Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large stencils that are in general not positive. We present an approach that yields stencils of minimal size, which are positive. We provide conditions on the point cloud geometry, so that positive stencils always exist. The new discretization method is compared to least squares approaches in terms of accuracy and computational performance.