  Virtual element methods is a new promising finite element methods using
general polygonal meshes. Its optimal a priori error estimates are well
established in the literature. In this paper, we take a different viewpoint. We
try to uncover the superconvergent property of the virtual element methods by
doing some local post-processing only on the degrees of freedom. Using linear
virtual element method as an example, we propose a universal recovery procedure
to improve the accuracy of gradient approximation for numerical methods using
general polygonal meshes. Its capability of serving as a posteriori error
estimators in adaptive methods is also investigated. Compared to the existing
residual-type a posteriori error estimators for the virtual element methods,
the recovery-type a posteriori error estimator based on the proposed gradient
recovery technique is much simpler in implementation and asymptotically exact.
A series of benchmark tests are presented to numerically illustrate the
superconvergence of recovered gradient and validate the asymptotical exactness
of the recovery-based a posteriori error estimator.
