Virtual Element Methods Without Extrinsic Stabilization

Abstract: 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 $L<sup>2$ projection of the gradient of virtual element functions is computable, and the $L<sup>2$ projector has a uniform lower bound on the gradient of virtual element function spaces in $L<sup>2$ norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.