Stabilization-Free H(curl) and H(div)-Conforming Virtual Element Method

Abstract: In this work, we propose a stabilization-free virtual element method for genreal order $\mathbf{H}(\operatorname{\mathbf{curl}})$ and $\mathbf{H}(\operatorname{div})$-conforming spaces. By the exact sequence of node, edge and face virtual element spaces, this method is applicable to PDEs involving $\nabla, \nabla\times$ and $\nabla\cdot$ operators. The key is to construct the noval serendipity virtual element spaces under the equivalence of the $L^2$-serendipity projector, from a sufficiently high order original space so that a stable polynomial projection is computable. The optimal approximation properties of the noval serendipity spaces are also proved.