Efficient spectral methods for the fourth-order elliptic eigenvalue problems

Abstract

An efficient spectral-Galerkin method for eigenvalue problems of the fourth-order elliptic equation on the unit ball is proposed in this paper. The efficiency of the method lies in the use of properly designed ball polynomials as basis functions. Error estimates for numerical eigenvalues and eigenvectors are conducted for the original fourth-order elliptic eigenvalue problem rather than the equivalent one-dimensional eigenvalue problem based on the pole condition in the literature. Numerical experiments are shown to demonstrate the efficiency of the algorithm and to validate the theoretical results.