Minimal positive stencils in meshfree finite difference methods for the Poisson equation
For researchers using meshfree methods for PDEs, this work solves the problem of non-positive stencils, which can cause instability, by providing a minimal and guaranteed positive stencil construction.
The paper presents a method for constructing minimal positive stencils in meshfree finite difference methods for the Poisson equation, ensuring all neighbor entries have the same sign. The approach yields stencils of minimal size and provides geometric conditions for their existence, outperforming least squares methods in accuracy and computational performance.
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.