  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.
