Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions
Immersed Virtual Element Methods for Electromagnetic Interface Problems in Three Dimensions
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DOI：
10.1142/s0218202523500112
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发表时间：
2022-02
影响因子：
3.5
通讯作者：
Shuhao Cao;Long Chen;Ruchi Guo
中科院分区：
文献类型：
--
作者：
Shuhao Cao;Long Chen;Ruchi Guo
用Maxweel型方程模拟电磁问题的有限元方法对近似空间的一致性非常敏感，而非一致性方法可能导致收敛损失。这一事实导致了几乎所有关于应用于电磁界面问题的文献中的界面不适应网格方法的根本障碍，因为它们是基于非协调空间的。本文提出了一种新的浸没虚拟元方法来求解三维$\mathbf{H}(\mathm{curl})$界面问题，其动机是将虚拟单元空间的整合性和浸入有限元空间的稳健逼近能力结合起来。该方法能够达到最优收敛。为了建立一个系统的框架，综合考虑了$H^1$、$\mathbf{H}(\mathm{curl})$和$\mathbf{H}(\mathm{div})$界面问题及其相应的面向问题的沉浸式虚拟元素空间。此外，还将建立de Rham复形，并在此基础上使用Hiptmair-Xu(HX)预处理器来开发$\mathbf{H}(\mathm{curl})$界面问题的快速求解器。
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.