We introduce the concept of half-closed nodes for nodal Discontinuous Galerkin (DG) discretisations. This is in contrast to more commonly used closed nodes in DG where in each element nodes are placed on every boundary. Half-closed nodes relax this constraint by only requiring nodes on a subset of the boundaries in each element, with this extra freedom in node placement allowing for increased efficiency in the assembly of DG operators. To determine which element boundaries half-closed nodes are placed on we outline a simple procedure based on switch functions. We examine the effect on operator sparsity from using the different types of nodes and show that in particular for the Laplace operator there is no difference in the sparsity from using half-closed or closed nodes. We also discuss in this work some linear solver techniques commonly used for Finite Element or Discontinuous Galerkin methods such as static condensation and block-based methods, and how they can be applied to half-closed DG discretisations.

翻译：我们提出了用于节点型间断伽辽金（DG）离散格式的半闭节点概念。与DG中更常用的闭节点（每个单元内所有边界均布置节点）不同，半闭节点放宽了这一约束：仅需在每个单元的部分边界上布置节点，这种节点布置的额外自由度可提升DG算子组装的效率。为确定各单元边界上布置半闭节点的具体位置，我们基于开关函数提出了一种简洁的流程。通过考察不同类型节点对算子稀疏性的影响，我们证明了对于拉普拉斯算子而言，使用半闭节点与闭节点的稀疏性没有差异。本文还讨论了有限元或间断伽辽金方法中常用的线性求解技术（如静态凝聚法和基于块的方法）在半闭DG离散格式中的适用性。