  This paper introduces bootstrap multigrid methods for solving eigenvalue
problems arising from the discretization of partial differential equations.
Inspired by the full bootstrap algebraic multigrid (BAMG) setup algorithm that
includes an AMG eigensolver, it is illustrated how the algorithm can be
simplified for the case of a discretized partial differential equation (PDE),
thereby developing a bootstrap geometric multigrid (BMG) approach. We
illustrate numerically the efficacy of the BMG method for: (1) recovering
eigenvalues having large multiplicity, (2) computing interior eigenvalues, and
(3) approximating shifted indefinite eigenvalue problems. Numerical experiments
are presented to illustrate the basic components and ideas behind the success
of the overall bootstrap multigrid approach. For completeness, we present a
simplified error analysis of a two-grid bootstrap algorithm for the
Laplace-Beltrami eigenvalue problem.
