Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local $H(\textrm{div})$-conforming macro finite element spaces such that the associated $L^2$ projection of the gradient of virtual element functions is computable, and the $L^2$ projector has a uniform lower bound on the gradient of virtual element function spaces in $L^2$ norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.

翻译：针对二阶椭圆问题，本文发展了无需求助于外部稳定化、适用于任意多项式次数的虚拟单元方法（VEMs），包括任意维度的非协调VEM和协调VEM。其关键在于构造局部$H(\textrm{div})$协调的宏观有限元空间，使得虚拟单元函数梯度的相关$L^2$投影可计算，并且$L^2$投影算子对虚拟单元函数空间在$L^2$范数下的梯度具有一致下界。本文推导了这些VEMs的最优误差估计，并通过数值实验验证了无外部稳定化VEMs的有效性。