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A virtual element method (VEM) with the first-order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual [Formula: see text]-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.
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This content will become publicly available on May 20, 2026
The VEM for time-harmonic Maxwell equations in inhomogeneous media with Lipschitz interface
It has been extensively studied in the literature that solving Maxwell equations is very sensitive to mesh structures, 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 broken spaces and non-conforming or polytopal meshes often used for inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces. This very issue can be potentially worsened by geometric singularities, making those methods particularly challenging to apply. In this paper, we present a lowest-order 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. We employ the “virtual mesh” technique originally invented in [S. Cao, L. Chen and R. Guo, A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh, Math. Models Methods Appl. Sci. 31 (2021) 2907–2936] for error analysis. This work admits three key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely [Formula: see text] regularity, [Formula: see text]; (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; (iii) we show that the stabilization term is needed to produce optimal convergent solutions for indefinite problems. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method.
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- Award ID(s):
- 2309777
- PAR ID:
- 10594876
- Publisher / Repository:
- World Scientific Publishing Company
- Date Published:
- Journal Name:
- Mathematical Models and Methods in Applied Sciences
- ISSN:
- 0218-2025
- Page Range / eLocation ID:
- 1 to 45
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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