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.

翻译：本文提出了一种基于约束最小二乘问题构建投影的实现通用虚拟元空间的框架。在有限元空间所采用的三元组结构基础上，我们引入了VEM元组的概念，该元组编码了构建这些投影所需的必要模块。利用该方法可定义多种虚拟元空间。本文讨论了H^k-协调空间（k=1,2）以及散度和旋度自由空间。该通用框架具有易于集成到现有有限元软件包中的优势，我们在开源软件包DUNE中进行了验证。