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. Several interesting phenomena for
approximating eigenvalues with high multiplicity are shown comparing
conventional multigrid ideas with the new bootstrap multigrid methods.