Efficient Spectral Collocation Method Based On Gauss Points For The Eigenvalue Problems

An effective spectral collocation method based on Gauss points is proposed for ellipse eigenvalue problems and transmission eigenvalue problems.Firstly,according to the orthogonal properties of Legendre polynomials,we construct a set of basis functions which satisfy the boundary conditions.Then,we expands the approximation solution by the set of basis functions.Secondly,using the three-item recursive relationship of the orthogonal polynomials,we compute the node values at each Gauss point for all the base functions and transform the discrete scheme into a linear eigen-system.Thirdly,by using the preconditioned iterative method,we can quickly compute the approximation eigenvalues and the corresponding eigenvectors.Finally,numerical experiments are given for the ellipse eigenvalue problems and the transmission eigenvalue problems respectively demonstrated by four examples for each.The numerical results show that the method is very effective.