A multigrid method for eigenvalue problem

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.