  We propose a multigrid correction scheme to solve a new Steklov eigenvalue
problem in inverse scattering. With this scheme, solving an eigenvalue problem
in a fine finite element space is reduced to solve a series of boundary value
problems in fine finite element spaces and a series of eigenvalue problems in
the coarsest finite element space. And the coefficient matrices associated with
those linear systems are constructed to be symmetric and positive definite. We
prove error estimates of eigenvalues and eigenfunctions. Numerical results
coincide in theoretical analysis and indicate our scheme is highly efficient in
solving the eigenvalue problem.
