  In this paper, an improved superconvergence analysis is presented for both
the Crouzeix-Raviart element and the Morley element. The main idea of the
analysis is to employ a discrete Helmholtz decomposition of the difference
between the canonical interpolation and the finite element solution for the
first order mixed Raviart--Thomas element and the mixed
Hellan--Herrmann--Johnson element, respectively. This, in particular, allows
for proving a full one order superconvergence result for these two mixed finite
elements. Finally, a full one order superconvergence result of both the
Crouzeix-Raviart element and the Morley element follows from their special
relations with the first order mixed Raviart--Thomas element and the mixed
Hellan--Herrmann--Johnson element respectively. Those superconvergence results
are also extended to mildly-structured meshes.
