Half-closed discontinuous Galerkin discretisations

TL;DR

This paper introduces half-closed nodes for discontinuous Galerkin discretisations, showing they maintain sparsity, improve assembly speed, and integrate well with existing solver techniques, offering a novel approach to DG methods.

Contribution

The paper proposes half-closed nodes for DG discretisations, demonstrating their advantages in sparsity, assembly speed, and solver integration, which is a new concept in DG methods.

Findings

No difference in Laplace operator sparsity pattern between closed and half-closed nodes.

Using Gauss-Radau points as half-closed nodes speeds up DG operator assembly.

Half-closed nodes can be effectively integrated with existing linear solver techniques.

Abstract

We introduce the concept of half-closed nodes for nodal discontinuous Galerkin (DG) discretisations. Unlike more commonly used closed nodes in DG, where on every element nodes are placed on all of its boundaries, half-closed nodes only require nodes to be placed on a subset of the element's boundaries. The effect of using different nodes on DG operator sparsity is studied and we find in particular for there to be no difference in the sparsity pattern of the Laplace operator whether closed or half-closed nodes are used. On quadrilateral/hexahedral elements we use the Gauss-Radau points as the half-closed nodes of choice, which we demonstrate is able to speed up DG operator assembly in addition to leverage previously known superconvergence results. 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.