A framework for implementing general virtual element spaces

TL;DR

This paper introduces a versatile framework for constructing and implementing general virtual element spaces using projections from constrained least squares problems, enabling easy integration into existing finite element software.

Contribution

It presents a novel VEM tuple concept for defining a wide range of virtual element spaces, including conforming and divergence-free spaces, with practical implementation in DUNE.

Findings

Framework can define various virtual element spaces

Easy integration into finite element software

Demonstrated implementation in DUNE

Abstract

In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a VEM tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss $H^k$-conforming spaces for $k=1,2$ as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package and we demonstrate this within the open source software package DUNE.