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A painless automatic hp-adaptive strategy for elliptic problems
Finite Elements in Analysis and Design ( IF 3.5 ) Pub Date : 2020-10-01 , DOI: 10.1016/j.finel.2020.103424
Vincent Darrigrand , David Pardo , Théophile Chaumont-Frelet , Ignacio Gómez-Revuelto , Luis Emilio Garcia-Castillo

In this work, we introduce a novel hp-adaptive strategy. The main goal is to minimize the complexity and implementational efforts hence increasing the robustness of the algorithm while keeping quasi-optimal results. We employ a multi-level hierarchical data structure imposing Dirichlet nodes to manage the so-called hanging nodes. The hp-adaptive strategy is based on performing quasi-optimal unrefinements. Taking advantage of the hierarchical structure of the basis functions both in terms of the element size h and the polynomial order of approximation p, we mark those with the lowest contributions to the energy of the solution and remove them. This straightforward unrefinement strategy does not require from a fine grid or complex data structures, making the algorithm flexible to many practical situations and existing implementations. On the other side, we also identify some limitations of the proposed strategy, namely: (a) data structures only support isotropic h-refinements (although p-anisotropic refinements are enabled), (b) we assume certain quasi-orthogonality properties of the basis functions in the energy norm, and (c) in this work, we restrict to symmetric and positive definite problems. We illustrate these and other advantages and limitations of the proposed hp-adaptive strategy with several one-and two-dimensional Poisson examples.

中文翻译:

一种解决椭圆问题的无痛自动 hp 自适应策略

在这项工作中,我们引入了一种新颖的 hp 自适应策略。主要目标是最小化复杂性和实现工作,从而提高算法的鲁棒性,同时保持准最佳结果。我们采用多级分层数据结构,强加 Dirichlet 节点来管理所谓的悬挂节点。hp 自适应策略基于执行准最佳未细化。利用基函数在元素大小 h 和近似多项式阶数 p 方面的层次结构,我们标记那些对解的能量贡献最低的那些并删除它们。这种直接的未细化策略不需要精细的网格或复杂的数据结构,使算法可以灵活地适应许多实际情况和现有实现。另一方面,我们还确定了所提出策略的一些局限性,即:(a)数据结构仅支持各向同性 h 细化(尽管启用了 p 各向异性细化),(b)我们假设了能量范数中的基函数,以及 (c) 在这项工作中,我们限制为对称和正定问题。我们通过几个一维和二维泊松示例说明了所提出的 hp 自适应策略的这些和其他优点和局限性。我们仅限于对称和正定问题。我们通过几个一维和二维泊松示例说明了所提出的 hp 自适应策略的这些和其他优点和局限性。我们仅限于对称和正定问题。我们通过几个一维和二维泊松示例说明了所提出的 hp 自适应策略的这些和其他优点和局限性。
更新日期:2020-10-01
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