In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally-adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, which ensure trace continuity on non-conforming meshes. Following this approach, we derive a finite element space that can be expressed as the original one plus well-defined linear constraints. Moreover, it requires minimum parallelization effort, using standard functionality available in existing large-scale finite element codes. Numerical experiments demonstrate its optimal mesh adaptation capability, robustness to cut location and parallel efficiency, on classical Poisson $hp$-adaptivity benchmarks. Our work opens the path to functional and geometrical error-driven dynamic mesh adaptation with the aggregated finite element method in large-scale realistic scenarios. Likewise, it can offer guidance for bridging other scalable unfitted methods and parallel adaptive mesh refinement.

中文翻译：

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基于并行树的自适应网格的聚合未拟合有限元方法

在这项工作中，我们提出了一种自适应未拟合有限元方案，它将聚合有限元方法与并行自适应网格细化相结合。我们在局部适应的笛卡尔森林树网格上引入了一种新颖的可扩展分布式内存实现方案。我们提出了一种两步算法来构建手头的有限元空间，该算法仔细混合了问题自由度的聚合约束，摆脱了小割单元问题，以及标准悬挂自由度约束，确保了非- 符合网格。按照这种方法，我们推导出一个有限元空间，它可以表示为原始空间加上明确定义的线性约束。此外，它需要最少的并行化工作，使用现有大型有限元代码中可用的标准功能。数值实验证明了其最佳网格自适应能力、对切割位置的鲁棒性和并行效率，在经典的泊松 $hp$-自适应基准上。我们的工作为在大规模现实场景中使用聚合有限元方法实现功能和几何误差驱动的动态网格自适应开辟了道路。同样，它可以为桥接其他可扩展的未拟合方法和并行自适应网格细化提供指导。我们的工作为在大规模现实场景中使用聚合有限元方法实现功能和几何误差驱动的动态网格自适应开辟了道路。同样，它可以为桥接其他可扩展的未拟合方法和并行自适应网格细化提供指导。我们的工作为在大规模现实场景中使用聚合有限元方法实现功能和几何误差驱动的动态网格自适应开辟了道路。同样，它可以为桥接其他可扩展的未拟合方法和并行自适应网格细化提供指导。