  In this paper, we present some enhanced error estimates for augmented
subspace methods with the nonconforming Crouzeix-Raviart (CR) element. Before
the novel estimates, we derive the explicit error estimates for the case of
single eigenpair and multiple eigenpairs based on our defined spectral
projection operators, respectively. Then we first strictly prove that the CR
element based augmented subspace method exhibits the second-order convergence
rate between the two steps of the augmented subspace iteration, which coincides
with the practical experimental results. The algebraic error estimates of
second order for the augmented subspace method explicitly elucidate the
dependence of the convergence rate of the algebraic error on the coarse space,
which provides new insights into the performance of the augmented subspace
method. Numerical experiments are finally supplied to verify these new estimate
results and the efficiency of our algorithms.
