AI 中文总结
本文针对泊松问题Cut有限元离散化的几何多重网格建立收敛理论，证明了W循环收敛界，通过数值实验分析了V循环、幽灵罚项及阈值对迭代次数的影响，为CutFEM的多重网格方法提供了理论与数值依据。
AI 中文摘要
我们为应用于泊松问题Cut有限元离散化的带顶点块光滑子的几何多重网格建立收敛理论。该框架处理非继承层级形式以及物理域与活动域的不匹配问题。利用离散延拓性质，我们证明了在网格尺寸和切割几何下一致的两级收敛界，以及在两级速率的附加小性假设下的W循环收敛界。数值实验特意使用更强的V循环，对此本文不主张任何收敛界。收敛常数随次数p增大而退化。降低幽灵罚项可减少迭代次数。对齐双单元模型呈现出阶为p⁻²的半定性阈值，而p次切割模式的可见性尺度呈指数下降。在模型阈值下的实验减少了迭代次数，但未建立组装算子阈值。
英文摘要
We develop a convergence theory for geometric multigrid with vertex-patch smoothers applied to cut finite element discretizations of the Poisson problem. The framework addresses non-inherited level forms and the mismatch between the physical and active domains. Using the discrete extension property, we prove two-level convergence bounds uniform in the mesh size and the cut geometry, and W-cycle bounds under an additional smallness assumption on the two-level rate. The numerical experiments intentionally use the stronger V-cycle, for which no convergence bound is claimed here. The convergence constants degrade with the degree $p$. Lowering the ghost penalty improves iteration counts. An aligned two-cell model exhibits a semidefiniteness threshold of order $p^{-2}$, whereas the visibility scale of a degree-$p$ cut mode decreases exponentially. Experiments at the model threshold reduce the iteration counts, but do not establish an assembled-operator threshold.
