In this paper, for the Steklov-Lamé eigenvalue problem, we propose a multigrid discretization scheme of discontinuous Galerkin method based on the shifted-inverse iteration. Based on the existing a priori error estimates, we give the error estimates for the proposed scheme and prove that the resulting approximations can achieve the optimal convergence order when the mesh sizes fit into some relationships. Finally, we combine the multigrid scheme and adaptive procedure to present some numerical examples which indicate that our scheme are locking-free and efficient for computing Steklov-Lamé eigenvalues.

We establish a finite element multigrid discretization scheme based on the shifted-inverse iteration for the Steklov-Lamé eigenproblem, and investigate the a posteriori error estimation of residual type for the scheme. Firstly, we give the error estimation of the approximate eigenfunction in the sense of L²(∂Ω) norm, then we give the a posteriori error indicators for the multigrid approximate solution, and prove the reliability and efficiency of the indicators. Finally, we use the a posteriori error indicators to design an adaptive multigrid algorithm for solving the Steklov-Lamé eigenproblem.