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Linking ghost penalty and aggregated unfitted methods
Computer Methods in Applied Mechanics and Engineering ( IF 7.2 ) Pub Date : 2021-11-09 , DOI: 10.1016/j.cma.2021.114232
Santiago Badia 1, 2 , Eric Neiva 2 , Francesc Verdugo 2
Affiliation  

In this work, we analyse the links between ghost penalty stabilisation and aggregation-based discrete extension operators for the numerical approximation of elliptic partial differential equations on unfitted meshes. We explore the behaviour of ghost penalty methods in the limit as the penalty parameter goes to infinity, which returns a strong version of these methods. We observe that these methods suffer locking in that limit. On the contrary, aggregated finite element spaces are locking-free because they can be expressed as an extension operator from well-posed to ill-posed degrees of freedom. Next, we propose novel ghost penalty methods that penalise the distance between the solution and its aggregation-based discrete extension. These methods are locking-free and converge to aggregated finite element methods in the infinite penalty parameter limit. We include an exhaustive set of numerical experiments in which we compare weak (ghost penalty) and strong (aggregated finite elements) schemes in terms of error quantities, condition numbers and sensitivity with respect to penalty coefficients on different geometries, intersection locations and mesh topologies.



中文翻译:

链接鬼惩罚和聚合的未拟合方法

在这项工作中,我们分析了重影惩罚稳定性和基于聚合的离散扩展算子之间的联系,以用于未拟合网格上椭圆偏微分方程的数值近似。当惩罚参数趋于无穷大时,我们探索了幽灵惩罚方法在极限中的行为,这将返回这些方法的强版本。我们观察到这些方法在该限制中受到锁定。相反,聚合有限元空间是无锁的,因为它们可以表示为从适定自由度到不适定自由度的扩展算子。接下来,我们提出了新的幽灵惩罚方法,惩罚解决方案与其基于聚合的离散扩展之间的距离。这些方法是无锁定的,并且在无限惩罚参数限制下收敛到聚合有限元方法。我们包括一组详尽的数值实验,其中我们在误差量、条件数和灵敏度方面比较弱(重影惩罚)和强(聚合有限元)方案,这些方案与不同几何形状、交叉位置和网格拓扑的惩罚系数有关。

更新日期:2021-11-10
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