A meshless Chebyshev collocation method for eigenvalue problems of the Helmholtz equation

In this paper, a meshless Chebyshev collocation method (CCM) for the numerical solution of the eigenvalue problems of the Helmholtz equation is presented. The Chebyshev polynomials are employed for the efficient and accurate approximation of the eigenfunctions to ensure the pseudo-spectral convergence of the CCM. Two different approaches, namely the generalized eigenvalue approach and the standard eigenvalue approach, which convert the original eigenvalue problem into the generalized eigenvalue problem and the standard eigenvalue problem respectively, are implemented. Five benchmark numerical examples are presented and discussed to demonstrate the accuracy and efficiency of the proposed CCM.