Stable Gaussian radial basis function method for solving Helmholtz equations

Abstract

‎Radial basis functions (RBFs) are a powerful tool for approximating the solution of high-dimensional problems‎. ‎They are often referred to as a meshfree method and can be spectrally accurate‎. ‎In this paper, we analyze a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion‎. ‎We develop our approach in two-dimensional spaces for solving Helmholtz equations‎. ‎In this paper, the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations‎. ‎Numerical examples are presented to demonstrate the effectiveness and robustness of the proposed method for solving two-dimensional Helmholtz equations‎.