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，发展了无外部稳定化且适用于任意多项式次数的虚拟单元方法，包括任意维度的非协调虚拟单元法和协调虚拟单元法。关键在于构造局部$H(\textrm{div})$协调宏单元有限元空间，使得虚拟单元函数梯度的关联$L^2$投影可计算，且该$L^2$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。推导了这些虚拟单元方法的最优误差估计，并通过数值实验验证了无外部稳定化虚拟单元方法的有效性。