Stabilization-Free H(curl) and H(div)-Conforming Virtual Element Method
Yuxuan Liao, Xue Feng, Yidong Huang
TL;DR
The paper addresses how to construct a stabilization-free VEM for general-order $\mathbf{H}(\operatorname{curl})$ and $\mathbf{H}(\operatorname{div})$ spaces by designing noval serendipity spaces that are equivalent, under an $L^2$-serendipity projector, to a higher-order original space so that a stable polynomial projection is computable. It extends the approach to 3D polyhedral meshes with an exact sequence linking $H^1$, $\mathbf{H}(\operatorname{curl})$, and $\mathbf{H}(\operatorname{div})$, preserving boundary continuity and B-compatibility and enabling PDEs involving $\nabla$, $\nabla\times$, and $\nabla\cdot$. The authors prove the optimal approximation properties of the serendipity spaces, establish the existence of a stable projection, and define an efficient, stabilization-free bilinear form through $a_h(\boldsymbol{u}_h,\boldsymbol{v}_h) = a_h(\boldsymbol{\Pi}_l\boldsymbol{u}_h, \boldsymbol{\Pi}_l\boldsymbol{v}_h)$. This framework supports general polyhedral meshes and general order spaces, broadening the applicability of VEM to complex geometries and multiphysics PDEs.
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.