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 under the mesh assumption that all the faces of each polytope are simplices. The key is to construct local $H({\rm div})$-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，本文发展了无需外在稳定化且可在任意多项式次数下实现的虚拟元方法，包括在任意维空间中基于多面体所有面均为单纯形的网格假设下的非协调虚拟元方法和协调虚拟元方法。其关键在于构造局部$H({\rm div})$协调的宏有限元空间，使得虚拟元函数梯度的关联$L^2$投影可计算，且该$L^2$投影子在虚拟元函数空间的$L^2$模梯度上具有一致下界。本文推导了这些虚拟元方法的最优误差估计，并通过数值实验验证了无需外在稳定化的虚拟元方法。