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.

翻译：在求解蜂巢六边形网格上的泊松方程时，我们证明$P_1$虚元在$H^1$范数下具有三阶超收敛性，在$L^2$和$L^\infty$范数下具有二阶超收敛性。我们定义了一种局部后处理方法，将超收敛的$P_1$解提升为最优阶逼近的$P_3$解。该理论通过数值实验得到了验证。