The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.

中文翻译：

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复杂几何体耦合体表面偏微分方程的未拟合 dG 方案

未拟合的不连续伽辽金 (UDG) 方法允许基于切割单元网格的偏微分方程 (PDE) 的保守 dG 离散化。因此，它特别适用于求解形状复杂的体域上的连续性方程。在本文基于和扩展第二作者的博士论文的基础上，我们展示了如何将该方法转移到曲面上的偏微分方程。受一类生物模型问题的启发，该问题包括静态体域及其表面上的连续性方程，我们为体表面模型提出了一种新的 UDG 方案。该方法结合了将表面偏微分方程扩展到更高维体域的思想与微量有限元方法的概念。特别关注保留离散偏微分方程守恒定律的离散类似物的必要步骤。通过使用几何的水平集表示来实现高度的几何灵活性。我们提出了理论结果来证明该方法的稳定性并研究其守恒性质。收敛以能量范数显示，数值结果显示体/表面 H 1 {H^{1}} - 和 L 2 {L^{2}} - 范数中的最佳收敛顺序。