  This paper is to introduce a type of full multigrid method for the nonlinear
eigenvalue problem. The main idea is to transform the solution of nonlinear
eigenvalue problem into a series of solutions of the corresponding linear
boundary value problems on the sequence of finite element spaces and nonlinear
eigenvalue problems on the coarsest finite element space. The linearized
boundary value problems are solved by some multigrid iterations. Besides the
multigrid iteration, all other efficient iteration methods for solving boundary
value problems can serve as the linear problem solver. We will prove that the
computational work of this new scheme is truly optimal, the same as solving the
linear corresponding boundary value problem. In this case, this type of
iteration scheme certainly improves the overfull efficiency of solving
nonlinear eigenvalue problems. Some numerical experiments are presented to
validate the efficiency of the new method.
