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:我们为嵌入$\mathbb{R ⁇ d$ 的表面的固定消化反应问题开发了一种新型的切片不连续的Galerkin(CutDG)方法。 CutDG方法基于将表层嵌入一个全维背景网格,并使用相关的不连续的片段单项顺序($k$)作为测试和试验功能。由于表层可以任意地穿透网格,我们设计了一种适当的稳定性,使我们能够利用强化的简化集成规范建立内向稳定、先验误差估计和条件编号估计。 由此产生的CutDG配方具有几何学般的坚固性, 也就是说,所有衍生的理论结果都与任何特定切割配置的常数无关。 数字示例支持了我们的理论发现。