Method Articles Efficient Algorithm and Convergence Analysis for Periodic Eigenvalue Problems Based on Two-Grid Discretization
Cai Zhou
*
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou, China.
Xu Xu
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou, China.
Qiuxia Tian
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou, China.
*Author to whom correspondence should be addressed.
Abstract
This paper proposes an efficient two-grid shifted inverse iteration algorithm for solving the eigenvalue problem of the Laplace operator with periodic boundary conditions:-Δω+ω=ξω. The method first obtains an initial approximate eigenpair (ξH,ωH) on a coarse grid, and then performs a one-step correction via shifted inverse iteration on a fine grid, maintaining high accuracy while significantly reducing computational cost. Theoretical analysis demonstrates that the numerical solutions achieve optimal convergence rates: the eigenfunction error in the energy norm is O(ϑh (ξt)), and the eigenvalue error is O(ϑ\(^2_h\)(ξt)), where ϑh (ξt) denotes the best approximation error of the t-th eigenfunction in the discrete finite element space. Numerical experiments validate the theoretical results and confirm the efficiency and optimal convergence performance of the algorithm for periodic eigenvalue problems.
Keywords: Eigenvalue problem, periodic boundary conditions, Two-grid method, shifted inverse iteration, finite element method, convergence analysis