Radial boundary elements method, a new approach on using radial basis functions to solve partial differential equations, efficiently

Conventionally, piecewise polynomials have been used in the boundary element method (BEM) to approximate unknown boundary values. However, since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for high dimensional domains, this paper proposes approximating the unknown values using RBFs. This new formulation is called the radial BEM. To calculate the singular boundary integrals in the radial BEM, the authors propose a new distribution of boundary source points that removes the singularity from the integrals. This allows the boundary integrals to be precisely calculated using the standard Gaussian quadrature rule with 16 quadrature nodes. Several numerical examples are presented to evaluate the efficiency of the radial BEM compared to standard BEM and RBF collocation method for solving partial differential equations (PDEs). The analytical and numerical studies demonstrate that the radial BEM is a superior version of BEM that will significantly enhance the application of BEM and RBFs in solving PDEs. © 2024 Elsevier Inc.