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Convergence of HX Preconditioner for Maxwell's Equations with Jump Coefficients (i): Various Extensions of The Regular Helmholtz Decomposition
arXiv - CS - Numerical Analysis Pub Date : 2017-08-19 , DOI: arxiv-1708.05850
Qiya Hu

This paper is the first one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu 2007) for Maxwell's equations with jump coefficients. In this paper we establish various extensions of the regular Helmholtz decomposition for edge finite element functions defined in three dimensional domains. The functions defined by the regular Helmholtz decompositions can preserve the zero tangential complement on faces and edges of polyhedral domains and some non-Lipchitz domains, and possess stability estimates with only a $logarithm$ factor. These regular Helmholtz decompositions will be used to prove convergence of the HX preconditioner for Maxwell's equations with jump coefficients in another paper (Hu 2017).

中文翻译:

具有跳跃系数的麦克斯韦方程组的 HX 预处理器收敛 (i):正则亥姆霍兹分解的各种扩展

本文是两篇系列文章中的第一篇,其目标是证明 HX Preconditioner(由 Hiptmair 和 Xu 2007 年提出)对具有跳跃系数的麦克斯韦方程组的收敛性。在本文中,我们为在三维域中定义的边缘有限元函数建立了常规亥姆霍兹分解的各种扩展。由正则亥姆霍兹分解定义的函数可以保留多面体域和一些非 Lipchitz 域的面和边上的零切向补数,并且仅具有对数因子的稳定性估计。在另一篇论文 (Hu 2017) 中,这些常规亥姆霍兹分解将用于证明具有跳跃系数的麦克斯韦方程组的 HX 预处理器的收敛性。
更新日期:2020-01-17
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