The spectral-Galerkin approximation of nonlinear eigenvalue problems

In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form −div(A∇u)+Vu+f(u2)u=λu,‖u‖L2=1. We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.