In this article, we develop and analyze a finite element method with the first family N\'ed\'elec elements of the lowest degree for solving a Maxwell interface problem modeled by a $\mathbf{H}(\text{curl})$-elliptic equation on unfitted meshes. To capture the jump conditions optimally, we construct and use $\mathbf{H}(\text{curl})$ immersed finite element (IFE) functions on interface elements while keep using the standard N\'ed\'elec functions on all the non-interface elements. We establish a few important properties for the IFE functions including the unisolvence according to the edge degrees of freedom, the exact sequence relating to the $H^1$ IFE functions and the optimal approximation capabilities. In order to achieve the optimal convergence rates, we employ a Petrov-Galerkin method in which the IFE functions are only used as the trial functions and the standard N\'ed\'elec functions are used as the test functions which can eliminate the non-conformity errors. We analyze the inf-sup conditions under certain conditions and show the optimal convergence rates which are also validated by numerical experiments.

中文翻译：

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在不拟合网格上求解具有最优收敛性的二维H（curl）-椭圆界面系统

在本文中，我们开发和分析一种有限元方法，该方法具有最低程度的第一个族N \'ed \'ele元素，用于解决由$ \ mathbf {H}（\ text {curl}）建模的Maxwell接口问题未拟合网格上的$椭圆方程。为了最佳地捕获跳转条件，我们在接口元素上构造并使用$ \ mathbf {H}（\ text {curl}）$浸入式有限元（IFE）函数，同时在所有接口上始终使用标准N \'ed \'elec函数非接口元素。我们为IFE函数建立了一些重要的属性，包括根据边缘自由度的均匀性，与$ H ^ 1 $ IFE函数有关的确切序列以及最佳逼近能力。为了达到最佳收敛速度，我们采用Petrov-Galerkin方法，其中IFE函数仅用作试验函数，而标准N \'ed \'elec函数用作测试函数，可以消除不符合项的错误。我们分析了在某些条件下的注入条件，并显示了最佳收敛速度，这也通过数值实验得到了验证。