A New Heterogeneous Multiscale Method for Time-Harmonic Maxwell's Equations
A New Heterogeneous Multiscale Method for Time-Harmonic Maxwell's Equations
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DOI：
10.1137/15m1039225
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发表时间：
2015-09
期刊：
影响因子：
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通讯作者：
P. Henning;Mario Ohlberger;B. Verfürth
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文献类型：
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作者：
P. Henning;Mario Ohlberger;B. Verfürth
本文提出了一种新的求解局部周期介质中时谐麦克斯韦方程组的非均匀多尺度方法。该方法是通过使用一个细胞问题的发散正则化。这使我们能够引入精细尺度校正器，不受繁琐的无发散约束，因此可以很容易地实现。为了分析该方法，我们首先回顾了时谐麦克斯韦方程组的经典均匀化理论，并利用双尺度均匀化方程中的散度正则化方法，推导出了一个新的均匀化结果.然后，我们表明，HMM是相当于这个方程的离散化。特别是，写在一个完全耦合的两个尺度制定的问题是一个相应的数值分析的方法的关键起点。通过这种方法，我们能够证明严格的先验误差估计在$\mathbf{H}（\mbox{curl}）$-和$H^{-1}$-范数，我们得到可靠和有效的本地化残差为基础的后验误差估计。
In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the $\mathbf{H}(\mbox{curl})$- and the $H^{-1}$-norm and we derive reliable and efficient localized residual-based a posteriori error estimates.