Asymptotic expansions of eigenvalues by both the Crouzeix–Raviart and enriched Crouzeix–Raviart elements

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and enriched Crouzeix–Raviart element. The optimal in sense that extrapolation based on them admit a fourth order convergence provided exact eigenfunctions smooth enough. major challenge establishing comes from fact canonical interpolation of nonconforming elements lacks crucial superclose property, nonconformity elements. main idea is to employ relation between lowest-order mixed Raviart–Thomas two elements, consequently make use property To overcome difficulty caused nonconformity, commuting operators further used, which turns consistency error problem into an problem. Then, series new results obtained show final expansions.