Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis

TL;DR

This work develops a meshfree Gaussian RBF collocation method for the fractional Poisson equation with extended Dirichlet data, leveraging a hypergeometric representation of the fractional Laplacian to achieve dimension-independent stiffness assembly and a Toeplitz structure under uniform grids. The authors prove rigorous stability and convergence results by coupling the shape parameter ε to the mesh size h through γ=(εh)^2 and deriving a fractional Poincaré inequality, yielding error bounds that separate a convergent component from a saturation term. Numerical experiments in 1D and 2D confirm the theoretical predictions, show improved conditioning relative to prior RBF approaches, and illustrate how the solver behaves under homogeneous and nonhomogeneous boundary conditions with guidance on selecting γ via c^*=εh. The methodology enables FFT-accelerated matrix-vector products and provides a first rigorous analytical treatment of RBF-based collocation methods for fractional PDEs, with practical implications for efficiently solving nonlocal models in higher dimensions.

Abstract

The paper introduces a new meshfree pseudospectral method based on Gaussian radial basis functions (RBFs) collocation to solve fractional Poisson equations. Hypergeometric functions are used to represent the fractional Laplacian of Gaussian RBFs, enabling an efficient computation of stiffness matrix entries. Unlike existing RBF-based methods, our approach ensures a Toeplitz structure in the stiffness matrix with equally spaced RBF centers, enabling efficient matrix-vector multiplications using fast Fourier transforms. We conduct a comprehensive study on the shape parameter selection, addressing challenges related to ill-conditioning and numerical stability. The main contribution of our work includes rigorous stability analysis and error estimates of the Gaussian RBF collocation method, representing a first attempt at the rigorous analysis of RBF-based methods for fractional PDEs to the best of our knowledge. We conduct numerical experiments to validate our analysis and provide practical insights for implementation.

Figures (5)

• Figure 1: Numerical errors in solving Poisson problem (\ref{['Poisson']}) with exact solution $u(x) = [(1-x^2)_+]^s$, where $c^* = 1/2$ and thus $\varepsilon = (N+1)/4$.
• Figure 1: Coefficient of saturation errors (i.e. $|a_{\bm\alpha}^{(\bm\beta)}(x)|/{\bm\alpha}!$) in 1D cases. (a) ${\bm \beta} = 0$; (b) ${\bm\beta} = 2$.
• Figure 2: Comparison of numerical errors for different $c^*$ in solving problem (\ref{['Poisson']}) with exact solution $u(x) = [(1-x^2)_+]^4$, where $\alpha = 1$ in (a)--(c), and $N = 127$ in (d).
• Figure 2: Coefficient of saturation errors at $x = 0.25$ in 1D cases.
• Figure 3: Comparison of numerical errors for different $c^*$ in solving problem (\ref{['Poisson']}) with $\alpha = 1$ on a unit disk, where the exact solution $u(\bm x) = [(1-|\bm x|^2)_+]^4$.

Theorems & Definitions (16)

• Lemma 1: approximation errors
• Proof 1
• Theorem 2: consistency error
• Proof 2
• Proposition 3
• Lemma 4
• Proof 3
• Proof 4: Proof of \ref{['prop:poincare']}
• Lemma 5
• Proof 5