this talk introduces a two-grid and a bootstrap multigrid finite element approximations to the laplace-beltrami eigenvalue problem on closed surfaces. the latter can be viewed as a special case of the bamg (bootstrap algebraic multi-grid) framework applying on surface finite element method. nonlinear eigenvalue problems are solved in the enriched finite element space on coarse mesh, while on the fine mesh only linear problems are approximated. several interesting phenomena for approximating eigenvalues with high multiplicity are shown comparing conventional two-grid/multigrid ideas with the new bootstrap multigrid methods. then some a posteriori error estimation technique for the multigrid iterate will be discussed which considers how accurate the linear problems need to be approximated to guarantee the overall optimal rate of convergence.