Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a 3D $\mathbf{H}(\mathrm{curl})$ interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the $H^1$, $\mathbf{H}(\mathrm{curl})$ and $\mathbf{H}(\mathrm{div})$ interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair-Xu (HX) preconditioner can be used to develop a fast solver for the $\mathbf{H}(\mathrm{curl})$ interface problem.

翻译：由Maxwell型方程建模的电磁问题的有限元方法对逼近空间的一致性高度敏感，非协调方法可能导致收敛性丧失。这一事实使得文献中几乎所有基于非协调空间的界面非拟合网格方法在应用于电磁界面问题时面临本质障碍。本文提出了一种新型浸入式虚拟单元方法用于求解三维$\mathbf{H}(\mathrm{curl})$界面问题，其动机在于结合虚拟单元空间的协调性与浸入式有限元空间的鲁棒逼近能力。该方法能够实现最优收敛。为建立系统性框架，本文统一考虑了$H^1$、$\mathbf{H}(\mathrm{curl})$和$\mathbf{H}(\mathrm{div})$界面问题及其对应的面向问题的浸入式虚拟单元空间。此外，将建立de Rham复形，在此基础上可运用Hiptmair-Xu (HX)预处理器为$\mathbf{H}(\mathrm{curl})$界面问题开发快速求解器。