SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 3004-3038, January 2020.
In bounded domains, the regularity of the solutions to boundary value problems depends on the geometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is, to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with Nédélec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

中文翻译：

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关于边缘有限元对电磁场的逼近。第3部分：对系数的敏感性

2020年1月，SIAM数学分析期刊，第52卷，第3期，第3004-3038页。
在有界域中，对边值问题的解决方案的规律性取决于几何形状以及进入模型定义的系数。时谐麦克斯韦方程组的情况尤其如此，其解是电磁场。在本文中，重点放在电场上。我们研究分数阶Sobolev空间$ H ^ s $，$ s \ in [0,1] $的正则性。精确地，我们的第一个目标是确定电场及其卷曲的规则性，即找到一些规则指数$ \ tau \ in（0,1）$，使它们都属于$ H ^ s $ ，对于所有$ s \ in [0，\ tau）$。之后，可以得出误差估计。在这里，误差定义为精确场与其近似值之间的差，后者由Nédélec'建立。的第一个有限元族。除了规则指数之外，还需要导出一个稳定常数，该常数将错误的范数与数据的范数相关联：这是我们的第二个目标。我们为正则指数和相对于系数的稳定常数提供了明确的表达式。我们还将讨论这些表达式的准确性，并提供一些数字插图。