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.

我们研究了应用于具有可重入面的网格上的对流反应方程的不连续伽辽金 (DG) 方法的收敛性。在这样的网格上，逆风数值通量不是平滑的，因此所得面项的数值积分只能预期为一阶准确。尽管存在这种不精确的积分，我们证明 DG 方法收敛于阶数$\mathcal{哦}(H^{磷+1/2})$，这与精确积分情况下的速率相同。因此，高阶精度不需要精确积分非平滑数值通量的专用正交规则。这些结果在线性平流和离散纵坐标输运方程的例子上得到了数值证实。