论文标题
并行通用$ hp $ - 自适应有限元软件的算法
Algorithms for Parallel Generic $hp$-adaptive Finite Element Software
论文作者
论文摘要
$ hp $ - 适应有限元方法（fem） - 一个独立选择的网格大小（$ h $）和多项式学位（$ p $）要在每个单元格上使用 - 长期以来一直在理论收敛属性比$ h $或$ h $或$ p $ aptaptive方法更好。但是，至少在某些部分的基础算法和缺乏可广泛使用的实现的困难之下，它并不广泛使用。当与连续有限元素一起使用时，尤其如此。 本文中，我们讨论了在分布式记忆，并行计算机上对$ hp $ - 适用性有限元方法进行全面且通用实现所必需的算法。特别是，我们将提出一种多阶段算法，用于适用于连续有限元空间的独特枚举（DOFS），描述加权负载平衡的考虑因素，并讨论在过程之间可变大小数据的传递。我们通过数值示例说明了算法的性能，并证明它们合理地扩展到至少16,384个消息传递接口（MPI）过程。 作为开源库Deal.ii的一部分，我们提供了算法的参考实现。
The $hp$-adaptive finite element method (FEM) - where one independently chooses the mesh size ($h$) and polynomial degree ($p$) to be used on each cell - has long been known to have better theoretical convergence properties than either $h$- or $p$-adaptive methods alone. However, it is not widely used, owing at least in parts to the difficulty of the underlying algorithms and the lack of widely usable implementations. This is particularly true when used with continuous finite elements. Herein, we discuss algorithms that are necessary for a comprehensive and generic implementation of $hp$-adaptive finite element methods on distributed-memory, parallel machines. In particular, we will present a multi-stage algorithm for the unique enumeration of degrees of freedom (DoFs) suitable for continuous finite element spaces, describe considerations for weighted load balancing, and discuss the transfer of variable size data between processes. We illustrate the performance of our algorithms with numerical examples, and demonstrate that they scale reasonably up to at least 16,384 Message Passing Interface (MPI) processes. We provide a reference implementation of our algorithms as part of the open-source library deal.II.
