Accurate Stabilization Techniques for RBF-FD Meshless Discretizations with Neumann Boundary Conditions

TL;DR

This paper addresses the challenge of accurately applying RBF-FD meshless methods to boundary value problems with Neumann boundary conditions, proposing stabilization techniques to improve stability and accuracy.

Contribution

The paper introduces new stabilization approaches for RBF-FD methods that mitigate ill-conditioning issues caused by Neumann boundary conditions.

Findings

Stability of RBF-FD improved with proposed techniques

Effective application to Helmholtz-Hodge decomposition demonstrated

Theoretical analysis links boundary normals to matrix conditioning

Abstract

A major obstacle to the application of the standard Radial Basis Function-generated Finite Difference (RBF-FD) meshless method is constituted by its inability to accurately and consistently solve boundary value problems involving Neumann boundary conditions (BCs). This is also due to ill-conditioning issues affecting the interpolation matrix when boundary derivatives are imposed in strong form. In this paper these ill-conditioning issues and subsequent instabilities affecting the application of the RBF-FD method in presence of Neumann BCs are analyzed both theoretically and numerically. The theoretical motivations for the onset of such issues are derived by highlighting the dependence of the determinant of the local interpolation matrix upon the boundary normals. Qualitative investigations are also carried out numerically by studying a reference stencil and looking for correlations between its geometry and the properties of the associated interpolation matrix. Based on the previous analyses, two approaches are derived to overcome the initial problem. The corresponding stabilization properties are finally assessed by succesfully applying such approaches to the stabilization of the Helmholtz-Hodge decomposition.