Asian Journal of Mathematics and Computer Research

This paper presents a reliable and efficient a posteriori error indicator for the eigenvalue problem of the Laplace operator subject to periodic boundary conditions. The proposed indicator decomposes the a posteriori error into element interior residuals and facet residuals. Theoretical analysis establishes that the indicator attains the optimal convergence order. Numerical experiments show that the four indicators λ₁,λ₂,λ₆,λ₁₀ all exhibit second-order O(h²) convergence. Among them, the λ₁​ indicator yields the smallest error over the full range of mesh sizes, demonstrating a distinct advantage in accuracy. The results provide numerical support for the selection of a posteriori error indicators and convergence verification in adaptive finite element methods for eigenvalue problems, and validate the theoretical reliability of a posteriori error estimates in adaptive mesh refinement.