A framework for implementing general virtual element spaces

In this paper we develop a framework for the construction and implementation of general virtual element spaces based on projections built from constraint least squares problems. We introduce the concept of a VEM tuple which encodes all the necessary building blocks to formulate the projections. Using this generic approach we present a wide range of virtual element spaces with additional properties. We showcase this approach with examples and build $H^k$-conforming spaces for $k=1,2$ as well as divergence and curl free spaces. The framework has the advantage of being easily integrated into any existing finite element package and we demonstrate this within the DUNE framework.