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:Maxwell方程模拟的电磁问题的有限元方法对近似空间的一致性非常敏感,不一致的方法可能会导致收敛的丢失。这一事实使得几乎所有文献中的界面不符合网格方法在应用于电磁界面问题时存在本质障碍,因为它们基于不一致的空间。本文提出了一种新颖的Immersed虚拟元方法来解决三维$\mathbf{H}(curl)$界面问题, 其动机是将虚拟元空间的一致性和Immersed有限元空间的强健逼近能力相结合。该方法能够达到最优收敛。为了开发一个系统的框架,考虑了$H^1,\mathbf{H}(curl)$和$\mathbf{H}(div)$界面问题及其对应的面向问题的Immersion虚拟元空间。此外,将建立de Rham复形,基于该复形可以使用Hiptmair-Xu(HX)预处理器来开发$\mathbf{H}(curl)$界面问题的快速求解器。