A short note on the accuracy of the discontinuous Galerkin method with reentrant faces

TL;DR

This paper proves that the discontinuous Galerkin method maintains high-order convergence on meshes with reentrant faces despite inexact face integration, supported by numerical examples.

Contribution

It establishes that DG methods achieve optimal convergence rates even with non-smooth flux integration on complex meshes, removing the need for specialized quadrature.

Findings

DG converges with order O(h^{p+1/2}) on reentrant meshes

Exact flux integration is not necessary for high-order accuracy

Numerical experiments confirm theoretical convergence rates

Abstract

We study the convergence of the discontinuous Galerkin (DG) method applied to the advection-reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order $\mathcal{O}(h^{p+1/2})$, which is the same rate as in the case of exact integration. Consequently, specialized quadrature rules that accurately integrate the non-smooth numerical fluxes are not required for high-order accuracy. These results are numerically corroborated on examples of linear advection and discrete ordinates transport equations.