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：我们提出了一种新型的切割间断伽辽金（CutDG）方法，用于处理嵌入$\mathbb{R}^d$中曲面上的稳态对流反应问题。CutDG方法基于将曲面嵌入到全维背景网格中，并使用阶数为$k$的关联间断分片多项式作为测试与试函数。由于曲面可能以任意方式切割网格，我们设计了一种合适的稳定化方案，使得能够利用增广的流线扩散范数建立inf-sup稳定性、先验误差估计以及条件数估计。所得到的CutDG公式在几何上是稳健的，即所有导出的理论结果中的常数均与特定的切割构型无关。数值算例支持了我们的理论发现。