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

TL;DR

This paper introduces a novel stabilization-free virtual element method for H(curl) and H(div) spaces, enabling stable and accurate solutions for PDEs involving curl and divergence operators without additional stabilization.

Contribution

The work develops a new serendipity virtual element space construction that is stabilization-free and maintains optimal approximation properties for H(curl) and H(div) conforming methods.

Findings

Constructed stabilization-free virtual element spaces.

Proved optimal approximation properties.

Applicable to PDEs with curl and divergence operators.

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.