  Two asymptotically exact a posteriori error estimates are proposed for
eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix--
Raviart elements. The main challenge in the design of such error estimators
comes from the nonconformity of the finite element spaces used. Such
nonconformity causes two difficulties, the first one is the construction of
high accuracy gradient recovery algorithms, the second one is a computable high
accuracy approximation of a consistency error term. The first difficulty was
solved for both nonconforming elements in a previous paper. Two methods are
proposed to solve the second difficulty in the present paper. In particular,
this allows the use of high accuracy gradient recovery techniques. Further, a
post-processing algorithm is designed by utilizing asymptotically exact a
posteriori error estimators to construct the weights of a combination of two
approximate eigenvalues. This algorithm requires to solve only one eigenvalue
problem and admits high accuracy eigenvalue approximations both theoretically
and numerically.
