Applications of Domain Decomposition and Partition of Unity Methods in Physics and Geometry
Applications of Domain Decomposition and Partition of Unity Methods in Physics and Geometry
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发表时间：
2010-01
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通讯作者：
M. Holst
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作者：
M. Holst
我们考虑一类自适应多级域分解类算法，该算法由自适应多级有限元、域分解和统一划分方法的组合构建。这些算法具有一些有趣的特征，例如非常低的通信要求，并且它们从统一方法的划分继承了简单而优雅的近似理论框架。它们也非常容易与高度复杂的顺序自适应有限元软件包一起使用，几乎不需要修改底层顺序有限元软件。并行算法可以实现为一个简单的循环，它同时在一组处理器上启动顺序局部自适应求解。我们首先回顾了Babuvska和Melenk的统一划分方法（PUM），并概述了PUM近似理论框架。然后，我们描述了一种变体，我们在此称为并行统一划分方法（PPUM），它是统一划分方法与 Bank 和 Holst 并行自适应算法的组合。然后，我们通过利用其继承的 PUM 分析框架，并采用 Xu 和 Zhou 最近的一些局部估计，得出 PPUM 的两个全局误差估计。然后，我们讨论基于对偶性的 PPUM 变体，它更适合某些应用，并推导出 PPUM 近似理论框架的合适变体。描述了我们使用 FETK 和 MC 软件包实现 PPUM 型算法。然后，我们提出一个简短的数值示例，涉及引力波模型中出现的爱因斯坦约束。
We consider a class of adaptive multilevel domain decomposition-like algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting features such as very low communication requirements, and they inherit a simple and elegant approximation theory framework from partition of unity methods. They are also very easy to use with highly complex sequential adaptive finite element packages, requiring little or no modification of the underlying sequential finite element software. The parallel algorithm can be implemented as a simple loop which starts off a sequential local adaptive solve on a collection of processors simultaneously. We first review the Partition of Unity Method (PUM) of Babuvska and Melenk, and outline the PUM approximation theory framework. We then describe a variant we refer to here as the Parallel Partition of Unity Method (PPUM), which is a combination of the Partition of Unity Method with the parallel adaptive algorithm of Bank and Holst. We then derive two global error estimates for PPUM, by exploiting the PUM analysis framework it inherits, and by employing some recent local estimates of Xu and Zhou. We then discuss a duality-based variant of PPUM which is more appropriate for certain applications, and we derive a suitable variant of the PPUM approximation theory framework. Our implementation of PPUM-type algorithms using the FETK and MC software packages is described. We then present a short numerical example involving the Einstein constraints arising in gravitational wave models.