It has been extensively studied in the literature that solving Maxwell equations is very sensitive to the mesh structure, space conformity and solution regularity. Roughly speaking, for almost all the methods in the literature, optimal convergence for low-regularity solutions heavily relies on conforming spaces and highly-regular simplicial meshes. This can be a significant limitation for many popular methods based on polytopal meshes in the case of inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces, and potentially worsened by geometric singularities, making many popular methods based on broken spaces, non-conforming or polytopal meshes particularly challenging to apply. In this article, we present a virtual element method for solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media with quite arbitrary polytopal meshes, and the media interface is allowed to have geometric singularity to cause low regularity. There are two key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely $\mathbf{H}^{\theta}$ regularity, $\theta\in(1/2,1]$; (ii) the polytopal element shape can be highly anisotropic and shrinking, and an explicit formula is established to describe the relationship between the shape regularity and solution regularity. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method.

翻译：文献中已广泛研究表明，求解Maxwell方程对网格结构、空间协调性和解的正则性极为敏感。粗略而言，文献中几乎所有方法对低正则解的最优收敛性都高度依赖于协调空间和高正则单纯形网格。对于基于多面体网格的众多流行方法而言，这可能在非均匀介质情形下构成显著局限——电磁参数的不连续性会导致介质界面附近的解正则性极低，且几何奇异性可能进一步恶化这一问题，使得许多基于破碎空间、非协调或多面体网格的流行方法特别难以应用。本文针对二维非均匀介质中具有相当任意多面体网格的待定时间谐波Maxwell方程，提出一种虚拟单元法，允许介质界面存在导致低正则性的几何奇异性。本研究有两项关键创新：（i）所提方法理论上能保证对仅具有$\mathbf{H}^{\theta}$正则性（$\theta∈(1/2,1]$）的解实现鲁棒最优收敛；（ii）多面体单元形状可呈现高度各向异性与收缩特性，并建立了描述形状正则性与解正则性关系的显式公式。大量数值实验将验证所提方法的有效性。