On a multigrid method for solving partial eigenproblems

Recently the direct application of a multigrid technique for computing the smallest eigenvalue and its corresponding eigenvector of a large symmetric positive definite matrix $A$ has been investigated in [5]. This method solves the eigenvalue problems on a sequence of nested grids using an interpolant of solution on each grid as initial guess for the next one and improving it by the full approximation scheme applied as an inner nonlinear multigrid method.
In the present paper, the generalization of the method for computing a few smallest eigenvalues and their corresponding eigenvectors of the elliptic self adjoint operator is presented. Moreover, the quality of the method is improved by using the nonlinear Gauss–Seidel iteration instead of its linearized version as pre- and post-smoothing steps. Finally, we give some advice for a good choice of multigrid-related parameters.