  A full multigrid finite element method is proposed for semilinear elliptic
equations. The main idea is to transform the solution of the semilinear problem
into a series of solutions of the corresponding linear boundary value problems
on the sequence of finite element spaces and semilinear problems on a very low
dimensional space. The linearized boundary value problems are solved by some
multigrid iterations. Besides the multigrid iteration, all other efficient
numerical methods can also serve as the linear solver for solving boundary
value problems. The optimality of the computational work is also proved.
Compared with the existing multigrid methods which need the bounded second
order derivatives of the nonlinear term, the proposed method only needs the
Lipschitz continuation in some sense of the nonlinear term.
