Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on non-conforming approximation spaces, which may cause a loss of accuracy for solving Maxwell equations, and the widely-used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing N\'ed\'elec-type immersed finite element spaces with a Petrov-Galerkin scheme that is able to produce optimal-convergent solutions. To establish a systematic framework, we analyze all the $H^1$, $\mathbf{H}(\text{curl})$ and $\mathbf{H}(\text{div})$ IFE spaces and form a discrete de Rham complex. Based on these fundamental results, we further develop a fast solver using a modified Hiptmair-Xu preconditioner which works for both the GMRES and CG methods.

翻译：Maxwell 界面问题在许多电磁应用中非常重要。 在 3D 计算中, 未经安装的网格方法特别具有吸引力, 因为它们可以绕过生成复杂的 3D 接口适合的网格。 然而, 许多不合适的网格方法依赖于不兼容的近似空格, 这可能造成解决 Maxwell 方程式的准确性损失, 而文献中广泛使用的处罚技术也许无助于恢复最佳趋同。 在文章中, 我们通过开发 N\'ed\' eleec 类型浸入的有限元素空间来提供补救。 基于这些基本结果, 我们进一步开发一个快速解答器, 使用经过修改的 hiptair- Xu 预设器, 用于 GMRES 和 CG 方法。