Two-grid methods for a class of nonlinear elliptic eigenvalue problems

In this article, we introduce and analyse some two-grid methods for nonlinear elliptic eigenvalue problems of the form -div(D del u) + Vu + f (u(2))u = lambda u, parallel to u parallel to(L2) = 1. We provide a priori error estimates for the ground state energy, the eigenvalue lambda and the eigenfunction u, in various Sobolev norms. We focus in particular on the Fourier spectral approximation (for periodic boundary conditions), and on the P-1 and P-2 finite element discretizations (for homogeneous Dirichlet boundary conditions), taking numerical integration errors into account. Finally, we provide numerical examples illustrating our analysis.