  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.
