SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A2620-A2654, January 2020.
We develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in $\mathbb{R}^d$ where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.

中文翻译：

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对流反应问题的稳定割断点间断Galerkin方法

SIAM科学计算杂志，第42卷，第5期，第A2620-A2654页，2020年1月。
我们开发对流反应问题的新型稳定割断面不连续Galerkin方法。感兴趣的域嵌入到$ \ mathbb {R} ^ d $中的结构化，不拟合的背景网格中，其中域边界可以任意方式切入网格。为了应对由小切角元素引起的鲁棒性问题，我们在嵌入边界附近引入幻影惩罚，以稳定与对流和反作用算子相关的某些（半）范数。确定了一些关于重影惩罚的抽象假设，使我们能够得出几何平稳且最优的先验误差和稳态平流-反应问题的条件数估计值，而不论特定的切割构型如何，该问题均成立。讨论了合适的虚假惩罚的可能实现。