Enhancing RBF-FD Efficiency for Highly Non-Uniform Node Distributions via Adaptivity

TL;DR

Nonuniform node distributions break traditional RBF-FD convergence and waste computational effort. The authors propose an adaptive PHS+Poly RBF-FD that tunes local polynomial degree $p_c$ and stencil size $n_s$ according to local fill distance $h_{X,\Omega_c}$ to achieve a user-specified global convergence order $g$, while maintaining or improving accuracy on highly nonuniform grids. A local refinement rule $p_c \ge g \frac{\log h_{e,\Omega}}{\log h_{X,\Omega_c}} + k - 1$ guides the adaptation and a fixed stencil relation $n_s = 2 n_p + 1$ ensures stability and solvability. Numerical experiments across 2D/3D Poisson problems, heat equations, and elastic-wave models demonstrate that the adaptive method preserves the desired convergence, increases sparsity of the differentiation matrix, and yields substantial efficiency gains in regions of high node density. This approach enables high-order, scalable PDE solving on irregular node layouts with potential impact on large-scale simulations.

Abstract

Radial basis function generated finite-difference (RBF-FD) methods have recently gained popularity due to their flexibility with irregular node distributions. However, the convergence theories in the literature, when applied to nonuniform node distributions, require shrinking fill distance and do not take advantage of areas with high data density. Non-adaptive approach using same stencil size and degree of appended polynomial will have higher local accuracy at high density region, but has no effect on the overall order of convergence and could be a waste of computational power. This work proposes an adaptive RBF-FD method that utilizes the local data density to achieve a desirable order accuracy. By performing polynomial refinement and using adaptive stencil size based on data density, the adaptive RBF-FD method yields differentiation matrices with higher sparsity while achieving the same user-specified convergence order for nonuniform point distributions. This allows the method to better leverage regions with higher node density, maintaining both accuracy and efficiency compared to standard non-adaptive RBF-FD methods.

Figures (14)

• Figure 1: A prototype figure showing centers and local stencil.
• Figure 2: Node distribution under different mesh ratio with $N=4096$ quasi-uniform data points in unit square $[-1,1]^2$ with various mesh ratios: (a) $\rho_X \approx 1$, (b) $\rho_X\approx 10^{2}$, and (c) $\rho_X \approx 10^{5}$.
• Figure 3: The process of achieving a global order of convergence in an PHS+Poly RBF-FD formulation. The order of convergence $g$ input by the user. The algorithm decides the degree of polynomial to be augmented, which also gives the number of polynomial terms required ($n_p$) and consequently the stencil size ($n_s$).
• Figure 4: Example 1: in $\Omega = [-1,1]^2$, setting $N=2500$ (a) Halton points, and (b) nonuniform nodes with large mesh ratio by sine-transform .
• Figure 5: Example 1: results by adaptive PHS+Poly RBF-FD method for uniform and nonuniform node distribution by $N=900$, (a,c): Sparsity pattern and (b,d): eigenvalue spectra of the system matrix.