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Robust and scalable h-adaptive aggregated unfitted finite elements for interface elliptic problems
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2020-06-19 , DOI: arxiv-2006.11042
Eric Neiva and Santiago Badia

This work introduces a novel, fully robust and highly-scalable, $h$-adaptive aggregated unfitted finite element method for large-scale interface elliptic problems. The new method is based on a recent distributed-memory implementation of the aggregated finite element method atop a highly-scalable Cartesian forest-of-trees mesh engine. It follows the classical approach of weakly coupling nonmatching discretisations at the interface to model internal discontinuities at the interface. We propose a natural extension of a single-domain parallel cell aggregation scheme to problems with a finite number of interfaces; it straightforwardly leads to aggregated finite element spaces that have the structure of a Cartesian product. We demonstrate, through standard numerical analysis and exhaustive numerical experimentation on several complex Poisson and linear elasticity benchmarks, that the new technique enjoys the following properties: well-posedness, robustness to cut location and material contrast, optimal (h-adaptive) approximation properties, high scalability and easy implementation in large-scale finite element codes. As a result, the method offers great potential as a useful finite element solver for large-scale multiphase and multiphysics problems modelled by partial differential equations.

中文翻译:

用于接口椭圆问题的稳健且可扩展的 h 自适应聚合未拟合有限元

这项工作引入了一种新颖的、完全鲁棒的、高度可扩展的、$h$ 自适应的聚合未拟合有限元方法,用于解决大规模界面椭圆问题。新方法基于最近在高度可扩展的笛卡尔树森林网格引擎之上聚合有限元方法的分布式内存实现。它遵循在界面处弱耦合非匹配离散的经典方法,以模拟界面处的内部不连续性。我们建议将单域并行单元聚合方案自然扩展到具有有限数量接口的问题;它直接导致具有笛卡尔积结构的聚合有限元空间。我们证明,通过对几个复杂的泊松和线性弹性基准的标准数值分析和详尽的数值实验,新技术具有以下特性:适定性、切割位置和材料对比度的鲁棒性、最佳(h 自适应)近似特性、高可扩展性和易于在大型有限元代码中实现。因此,该方法作为一种有用的有限元求解器具有巨大的潜力,可以解决由偏微分方程建模的大规模多相和多物理场问题。
更新日期:2020-06-22
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