A Multigrid Shifted-Inverse Iteration Algorithm for Periodic Boundary Eigenvalue Problems and its Convergence Analysis

Abstract

This paper systematically investigates a multigrid finite element method and its convergence theory for a class of second-order Laplace eigenvalue problems subject to periodic boundary conditions. We establish the variational formulation of the eigenvalue problem in periodic Sobolev spaces, construct a periodic finite element discretization scheme, and derive the corresponding algebraic eigenvalue problem. Two shifted-inverse iteration algorithms based on multigrid discretizations are proposed: one based on Rayleigh quotient iteration and the other employing a fixed-shift strategy. Notably, compared with the Rayleigh quotient iteration, the fixed-shift strategy exhibits superior numerical stability with a slower-growing condition number. By introducing the solution operator framework and error measure tools, we analyze in detail the approximation properties of the discrete eigenvalues and eigenfunctions. Under suitable regularity conditions, we derive error estimates for the eigenfunctions in the energy norm and negative norm, as well as an upper bound for the eigenvalue error. Theoretical results indicate that the proposed multigrid algorithms achieve optimal convergence orders during mesh refinement, and the convergence is independent of the number of grid levels. Numerical analysis validates the effectiveness of the theoretical results and the robustness of the algorithms.