Some observations regarding the RBF-FD approximation accuracy dependence on stencil size

TL;DR

The paper investigates how the stencil size $n$ in Polyharmonic Splines with monomial augmentation-based RBF-FD affects Poisson-equation accuracy on scattered nodes, revealing oscillatory error behavior with distinct local minima. By analyzing the spatial distribution of signed pointwise errors and introducing the global indicator-like quantity $\\delta N^{\pm}$, the authors link minima in the global error to changes in error sign patterns across the domain. They perform extensive numerical experiments across discretization scales, node layouts, boundary conditions, domain shapes, dimensionalities, differential operators, and analytical solutions, showing the phenomenon is robust though problem-dependent in exact minima locations. A practical heatsink example demonstrates potential real-world relevance and highlights the trade-off between stencil size and discretization refinement. The work suggests a direction for selecting locally optimal stencils without increasing order or refinement, and outlines future work toward theoretical explanations, broader problem classes, and physically meaningful error indicators that do not rely on an exact solution.

Abstract

When solving partial differential equations on scattered nodes using the Radial Basis Function-generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the signed approximation error. Based on this observation we are able to introduce a numerical quantity that could indicate whether a given stencil size is locally optimal. This work is an extension of our ICCS 2023 conference paper.

Figures (15)

• Figure 1: Example discretisation set generated by the DIVG algorithm. Example stencils are also displayed.
• Figure 2: Dependence of the approximation error on the stencil size $n$. The data used to make this plot is available in the appendix.
• Figure 3: Spatial dependence of $e_\mathrm{poiss}^\pm$ in some local extrema. The colour scale is the same for all drawn plots.
• Figure 4: The quantities $\delta N^\pm (n)$ along with the spatial profiles of the signs of $e^\pm$ for some chosen stencil sizes. For convenience, $\delta N^\pm = 0$ is marked with a green line.
• Figure 5: Behaviour of the approximation errors under a refinement of the discretisation.