  In this paper, a novel multigrid method based on Newton iteration is proposed
to solve nonlinear eigenvalue problems. Instead of handling the eigenvalue
$\lambda$ and eigenfunction $u$ separately, we treat the eigenpair $(\lambda,
u)$ as one element in a product space $\mathbb R \times H_0^1(\Omega)$. Then in
the presented multigrid method, only one discrete linear boundary value problem
needs to be solved for each level of the multigrid sequence. Because we avoid
solving large-scale nonlinear eigenvalue problems directly, the overall
efficiency is significantly improved. The optimal error estimate and linear
computational complexity can be derived simultaneously. In addition, we also
provide an improved multigrid method coupled with a mixing scheme to further
guarantee the convergence and stability of the iteration scheme. More
importantly, we prove convergence for the residuals after each iteration step.
For nonlinear eigenvalue problems, such theoretical analysis is missing from
the existing literatures on the mixing iteration scheme.
