Superconvergent P1 honeycomb virtual elements and lifted P3 solutions

When solving the Poisson equation on honeycomb hexagonal grids, we show that the $P_1$ virtual element is three-order superconvergent in $H^1$-norm, and two-order superconvergent in $L^2$ and $L^\infty$ norms. We define a local post-process which lifts the superconvergent $P_1$ solution to a $P_3$ solution of the optimal-order approximation. The theory is confirmed by a numerical test.