A stable and high-order accurate D-RBF-PU method via hybrid kernels

Articles in Press

Document Type : Research Article

Authors

Mohsen Esmaeilbeigi ¹

Roberto Cavoretto ²

¹ Faculty of Mathematical and Statistical Sciences, Malayer University, Malayer, Iran.

² Department of Mathematics “Giuseppe Peano”, University of Torino, Torino, Italy.

10.22067/ijnao.2026.98442.1865

Abstract

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) method is a powerful mesh-free technique for solving Partial Differential Equations (PDEs). However, its performance is inherently constrained by a critical trade-off between accuracy and numerical stability. Finite-order smoothness kernels, such as Thin Plate Splines (TPS) and Mat{'e}rn, ensure well-conditioned systems but severely limit the convergence rate. Conversely, infinitely smooth kernels, such as Gaussians, yield high accuracy but suffer from severe ill-conditioning, leading to computational breakdown at finer discretizations. In this paper, we propose a novel approach using hybrid kernels, strategically combining Gaussian kernels with finitely smooth kernels to effectively bridge this gap.
Numerical experiments on 2D benchmark elliptic PDEs demonstrate that the proposed hybrid formulations achieve an optimal balance. Specifically, hybrid kernels exhibit robust, high-order convergence rates that remain highly competitive with, and in certain structural combinations significantly surpass, the accuracy of pure infinitely smooth bases. Simultaneously, they maintain manageable local and global condition numbers, allowing the method to successfully scale to finer spatial resolutions where traditional infinitely smooth kernels fail. These results confirm that the hybrid kernel approach significantly enhances the robustness and accuracy of the D-RBF-PU method, offering a highly practical solution for large-scale simulations.

Keywords

Radial Basis Functions

Partition of Unity

Hybrid Kernels

Accuracy-Stability Trade-Off

Subjects

• Computational Mathematics

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