H(div) and H(curl)-conforming VEM

TL;DR

This paper introduces new Virtual Element Spaces conforming to $H({\rm div})$ and $H({\rm \bf curl})$ on complex polygons and polyhedra, generalizing finite elements for PDE approximation.

Contribution

It develops $H({\rm div})$ and $H({\rm \bf curl})$-conforming VEM spaces on general meshes and discusses their use in PDE approximation and exact sequence construction.

Findings

Spaces applicable to general polygonal and polyhedral meshes

Framework for PDE approximation using these spaces

Construction of exact sequences of VEM spaces

Abstract

In the present paper we construct Virtual Element Spaces that are $H({\rm div})$-conforming and $H({\rm \bf curl})$-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known Finite Elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the construction of exact sequences of VEM spaces.