论文标题
在三个维度中浸入电磁界面问题的虚拟元素方法
Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions
论文作者
论文摘要
由Maxwell型方程建模的电磁问题的有限元方法对近似空间的一致性高度敏感，不合格方法可能会导致收敛损失。这一事实导致了文献中几乎所有界面未拟合的网格方法的基本障碍，内容涉及到电磁界面问题，因为它们基于不合格的空间。在这项工作中，开发了一种新颖的浸入虚拟元素方法，用于求解3D $ \ mathbf {h}（\ mathrm {curl}）$接口问题，动机是结合虚拟元素空间的合规性和可靠的有限元元素空间的稳健近似能力。所提出的方法能够实现最佳收敛。要开发系统的框架，$ h^1 $，$ \ mathbf {h}（\ mathrm {curl}）$和$ \ mathbf {h}（\ mathrm {div}）$接口问题及其相应的问题与面向问题的无味虚拟元素空间都被认为是所有的。此外，将基于hiptmair-XU（HX）预处理器来建立DE RHAM复合体，用于开发$ \ MathBf {H}（\ MathRM {Curl}）$接口问题的快速求解器。
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.
