We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection-reaction problems on surfaces embedded in $\mathbb{R}^d$. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order $k$ as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.

翻译：我们发展了一种新的切割不连续Galerkin（CutDG）方法用于 $\mathbb{R}^d$ 中嵌入曲面的静止平流反应问题。 CutDG 方法基于将曲面嵌入到全维背景网格中并使用相应的阶数为 $k$ 的不连续分段多项式作为测试和试验函数。由于曲面可以任意裁剪网格，因此我们设计了合适的稳定化方法，使用增强的流线扩散范数建立了基础的黎曼问题，得到了先验误差、条件数估计和inf-sup稳定性。得到的CutDG公式在几何上是鲁棒的，这意味着所有的推导理论结果都是独立于任何特定切割配置的常数。数值算例支持了我们的理论发现。