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

TL;DR

This paper introduces a novel radial boundary elements method (radial BEM) that employs smooth radial basis functions for more stable and accurate solutions of PDEs, with a new approach to handle boundary integrals efficiently.

Contribution

It proposes a new formulation of BEM using radial basis functions and a novel boundary source point distribution to accurately compute singular integrals.

Findings

Radial BEM outperforms standard BEM and RBF collocation in efficiency.

The new boundary source point distribution removes singularities from integrals.

Numerical examples confirm the high accuracy and stability of the proposed method.

Abstract

Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for high dimensional domains, the unknown values are approximated by the RBFs in this paper. Therefore, a new formulation of BEM, called radial BEM, is obtained. To calculate singular boundary integrals of the new method, we propose a new distribution for boundary source points that removes singularity from the integrals. Therefore, the boundary integrals are calculated precisely by the standard Gaussian quadrature rule (GQR) with n = 16 quadrature nodes. Several numerical examples are presented to check the efficiency of the radial BEM versus standard BEM and RBF collocation method for solving partial differential equations (PDEs). Analytical and numerical studies presented in this paper admit the radial BEM as a perfect version of BEM which will enrich the contribution of BEM and RBFs in solving PDEs, impressively.