A Framework for Implementing General Virtual Element Spaces

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B229-B253, June 2024.
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 virtual element method (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 [math]-conforming spaces for [math] 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.dune-project.org/dune-fem/dune-vem-paper and in the supplementary materials (128492_2_supp_546442_s3hsrj.zip [22KB]).