非均匀介质中低正则性时谐Maxwell方程虚元方法的各向异性分析
Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity
AI总结：
本文针对二维非均匀介质中含低正则性解的时谐Maxwell方程，提出一种适用于任意多面体网格的虚元方法，理论保证在$\mathbf{H}^θ$（$θ\in(1/2,1]$）正则性下实现鲁棒最优收敛，并建立单元形状正则性与解正则性之间的显式关系。
摘要：
文献中已有大量研究表明，求解Maxwell方程对网格结构、空间协调性和解的正则性非常敏感。粗略地说，对于文献中几乎所有的方法，低正则性解的最优收敛都严重依赖于协调空间和高正则单纯形网格。对于许多基于多面体网格的流行方法而言，这在非均匀介质情形下可能是一个显著限制，因为电磁参数的不连续性会导致介质界面附近解的正则性相当低，并可能因几何奇异性而进一步恶化，使得许多基于破碎空间、非协调或多面体网格的流行方法尤其难以应用。本文提出一种虚元方法，用于在具有相当任意多面体网格的二维非均匀介质中求解不定时谐Maxwell方程，并允许介质界面具有几何奇异性从而导致低正则性。本文有两个关键创新点：（i）所提方法在理论上被保证对仅具有$\mathbf{H}^θ$正则性（$θ\in(1/2,1]$）的解实现鲁棒的最优收敛；（ii）多面体单元形状可以是高度各向异性和收缩的，并建立了描述形状正则性与解正则性之间关系的显式公式。文中将给出大量数值实验以证明所提方法的有效性。
英文摘要：
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 $\mathbf{H}^θ$ regularity, $θ\in(1/2,1]$; (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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
