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：文献已广泛研究Maxwell方程求解对网格结构、空间协调性及解正则性的高度敏感性。简言之，现有方法中，低正则性解的最优收敛性几乎完全依赖于协调空间与高正则单形网格。对于基于多面体网格的诸多流行方法而言，非均匀介质情形下电磁参数的不连续性可能导致介质界面附近解的正则性极低，加之几何奇异性可能进一步恶化，使得基于破碎空间、非协调或多面体网格的诸多方法在应用中面临重大挑战。本文提出一种虚拟元方法，用于求解二维非均匀介质中具有相当任意多面体网格的不定时间谐波Maxwell方程，且允许介质界面存在导致低正则性的几何奇异性。本文有两项关键创新：(i) 所提方法理论上可保证对于仅具$\mathbf{H}^{\theta}$正则性（$\theta\in(1/2,1]$）的解实现鲁棒最优收敛；(ii) 多面体单元形状可高度各向异性甚至收缩，并建立了显式公式来描述形状正则性与解正则性之间的关系。大量数值实验将验证所提方法的有效性。