  In this paper, three high-accuracy methods for eigenvalues of second order
elliptic operators are proposed by using the nonconforming Crouzeix-Raviart(CR
for short) element and the nonconforming enriched Crouzeix-Raviart(ECR for
short) element. They are based on a crucial full one order superconvergence of
the first order mixed Raviart-Thomas(RT for short) element. The main ingredient
of such a superconvergence analysis is to employ a discrete Helmholtz
decomposition of the difference between the canonical interpolation and the
finite element solution of the RT element. In particular, it allows for some
vital cancellation between terms in one key sum of boundary terms.
Consequently, a full one order superconvergence follows from a special relation
between the CR element and the RT element, and the equivalence between the ECR
element and the RT element for these two nonconforming elements. These
superconvergence results improve those in literature from a half order to a
full one order for the RT element, the CR element and the ECR element. Based on
the aforementioned superconvergence of the RT element, asymptotic expansions of
eigenvalues are established and employed to achieve high accuracy extrapolation
methods for these two nonconforming elements. In contrast to a classic analysis
in literature, the novelty herein is to use not only the canonical
interpolations of these nonconforming elements but also that of the RT element
to analyze such asymptotic expansions. Based on the superconvergence of these
nonconforming elements, asymptotically exact a posteriori error estimators of
eigenvalues are constructed and analyzed for them. Finally, two post-processing
methods are proposed to improve accuracy of approximate eigenvalues by
employing these a posteriori error estimators.Numerical tests are provided to
justify and compare the performance of the aforementioned methods.
