H-RBFs collocation method for solving the Helmholtz equation with variable coefficients

Abstract

This study presents a hierarchical radial basis function (H-RBFs) collocation method for addressing the Helmholtz equation with variable coefficients. The new method is truely meshfree and is easy to implement. The trial spaces of the H-RBFs method are constructed employing successively refined scattered nodal sets as well as scaled, compactly supported radial basis functions (CSRBFs) characterized by distinct support radii. This strategy enables flexible construction of approximation spaces that support arbitrary dimensionality and adjustable smoothness, while circumventing the significant computational overhead associated with conventional mesh-based methods. Discretization in this framework only requires direct evaluation at collocation points, which greatly reduces the complexity associated with variational and integral operations. The proposed H-RBFs collocation method achieves improved accuracy and higher computational efficiency for scattered points over complex domains, and produces a discrete algebraic system with high sparsity. This characteristic enables the new method to improve computational efficiency greatly. Finally, these conclusions have been obtained through the application of direct numerical simulation.