Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
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DOI：
10.1090/mcom/3447
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发表时间：
2017-11
期刊：
影响因子：
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通讯作者：
M. Bonazzoli;V. Dolean;I. Graham;E. Spence;Pierre-Henri Tournier
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文献类型：
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作者：
M. Bonazzoli;V. Dolean;I. Graham;E. Spence;Pierre-Henri Tournier
本文严格地分析了具有吸收项的时谐麦克斯韦方程的预条件子，其中偏微分方程采用任意阶的卷曲协调有限元方法离散，预条件子采用加性施瓦茨区域分解方法构造。这里开发的理论表明，如果吸收是足够大的，如果子域和粗网格直径和重叠选择适当的，然后经典的两级重叠添加剂施瓦茨预处理（与PEC边界条件的子域）执行最佳-在这个意义上，GMRES收敛在一个波数独立的迭代次数-吸收的问题。该理论的一个重要特征是，它允许从低阶元素构建粗空间，即使PDE使用高阶元素离散化。它还表明具有最小重叠的加法方法可以是鲁棒的。数值实验说明了理论和它对各种参数的依赖。这些实验激发了一些扩展的预条件，具有更好的鲁棒性的问题，吸收较少，包括传播的情况下。在本文的最后，我们说明了这些在两个实质性的应用程序的性能;第一个（一个问题与吸收所产生的医学成像）显示的经验鲁棒性的预处理器对异质性，和第二（散射的眼镜蛇腔）显示良好的可扩展性的预处理器高达3,000个处理器。
This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.