In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed $L^2$-inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted $L^2$-inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904].

在本文中，我们分析了二阶椭圆问题的小面多面网格上的非相容虚拟单元方法。我们为 2D 和 3D 非一致性虚拟单元方法提出了新的稳定性形式。对于 2D 情况，稳定性形式由近似切向导数的内积和加权的内积之和定义。$L^2$-网格元素边界上某些投影的内积。对于 3D 情况，稳定性形式由加权定义$L^2$-网格元素边界上的内积。我们证明了配备这种稳定性形式的非相容虚拟单元方法的最佳收敛性。最后，提出了几个数值实验来验证我们的分析并将所提出的稳定形式与标准稳定形式的性能进行比较[B. Ayuso de Dios、K. Lipnikov 和 G. Manzini，非相容虚元法，ESAIM Math。模型。数字。肛门。 50（2016）879–904]。