The Shifted-Inverse Iteration Based on the Multigrid Discretizations for Eigenvalue Problems

Abstract

The shifted-inverse iteration based on the multigrid discretizations developed in recent years is an efficient computation method for eigenvalue problems. In this paper, for general self-adjoint eigenvalue problems, including the Maxwell eigenvalue problem and integral operator eigenvalue problem, we establish the inverse iteration with fixed shift based on multigrid discretizations. We study in depth the inverse iteration with fixed shift and Rayleigh quotient iteration based on multigrid discretizations and first prove under general conditions the error estimates and convergence for the iterative solution approximating the exact solution of the original eigenvalue problems, especially in an adaptive fashion. Finally, we present some numerical examples performed to validate our theoretical results.