A Numerical Study of Combining RBF Interpolation and Finite Differences to Approximate Differential Operators

TL;DR

This paper compares two RBF-based operator-approximation strategies on a 2D Poisson problem: the standard RBF-FD and a hybrid method that uses RBF interpolation to realize virtual FD stencils. By varying monomial augmentation degree $m$ and virtual-stencil spacing $\delta=\sigma h$ across five-point and nine-point stencils, the study reveals regimes where the hybrid approach can outperform RBF-FD, particularly near the optimal $\sigma$ when $m$ is modest, albeit with higher computational cost. The results highlight that tuning $\sigma$ and $m$ is crucial for maximizing accuracy, and that higher-order RBF-FD generally dominates the hybrid method for large $m$, while the hybrid shines at specific parameter settings for lower $m$. The work underscores the potential of hybrid meshless-FD schemes for adapting existing FD stencils to irregular node layouts, with practical implications for complex geometries and problem-specific discretizations.

Abstract

This paper focuses on RBF-based meshless methods for approximating differential operators, one of the most popular being RBF-FD. Recently, a hybrid approach was introduced that combines RBF interpolation and traditional finite difference stencils. We compare the accuracy of this method and RBF-FD on a two-dimensional Poisson problem for standard five-point and nine-point stencils and different method parameters.

Figures (6)

• Figure 1: An example of a scattered node set $X \subset \mathbb{R}^2$ on the left-hand side and a stencil with six nearest nodes to the central node on the right-hand side. For clarity, stencil nodes are additionally encircled.
• Figure 2: An example of a virtual stencil for a single point on the left-hand side and for all points from the domain on the right-hand side.
• Figure 3: Error comparison for 2nd order monomial augmentation and the five-point and nine-point stencil.
• Figure 4: Error comparison for 4th order monomial augmentation and the five-point and nine-point stencil.
• Figure 5: Error comparison for the alternative interpolation method.