A type of multilevel method for the Steklov eigenvalue problem

A new type of iteration method is proposed in this paper to solve the Steklov eigenvalue problem by the finite element method. In this scheme, solving the Steklov eigenvalue problem is transformed into a series of solutions of boundary value problems on multilevel meshes by the multigrid method and solutions of the Steklov eigenvalue problem on the coarsest mesh. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The computational work of this new scheme for the Steklov eigenvalue problem can reach the same optimal order as the solution of the corresponding boundary value problem. Therefore, an improvement of efficiency for the Steklov eigenvalue solving method can be achieved. Some numerical experiments are presented to validate the efficiency of the new method.