Stabilization-Free Virtual Element Methods

Stabilization-free virtual element methods in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming virtual element method in arbitrary dimension and a conforming virtual element method in two dimensions. 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 stabilization-free virtual element methods. Numerical results are provided to verify the convergence rates.