Half-closed discontinuous Galerkin discretisations

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.

Figures (21)

• Figure 1: From left to right, examples of $p=2$ closed, open and half-closed nodes on a 2D quadrilateral element.
• Figure 2: Example switch functions on a simplex and quadrilateral mesh in 2D. Switch function values are shown with $+,-$ for $+1,-1$ respectively. These switch functions are valid as the values along inter-element boundaries have opposing signs.
• Figure 3: Communication pattern for boundary terms between neighbouring elements in discrete divergence operator for different nodes. On the left for closed nodes only the nodal values on the boundary are needed to evaluate the boundary terms. In the middle for open nodes as no nodes are on the boundary all the nodal values in both elements are needed from both elements. On the right for half-closed nodes, in the left element only the nodal values on the boundary are needed whilst on the right as no nodes are on the boundary all the nodal values in the element are needed similar to the open case.
• Figure 4: Comparison sparsity of the divergence operator on 1D mesh with three $p=2$ elements and periodic boundary conditions. The flux here at every inter-element boundary is taken to be $\hat{F}(F_L, F_R) = F_R$, the value from the right element. From left to right, the sparsity pattern of the divergence with this numerical flux using closed, open, and half-closed nodes is shown.
• Figure 5: Effect of applying an inverse mass matrix (center matrix) on the right of an arbitrary operator (left matrix). In this example the operators correspond to the two element $p=1$ shown at the bottom. The operator initially has only one non-zero per column. After applying the mass matrix on the right this spreads the non-zero entries across the rows of each block (right matrix).