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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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Anisotropic Locally-Conformal Perfectly Matched Layer for Higher Order Curvilinear Finite-Element Modeling
A perfectly matched layer (PML) method is proposed for electrically large curvilinear meshes based on a higher order finite-element modeling paradigm and the concept of transformation electromagnetics. The method maps the non-Maxwellian formulation of the locally conformal PML to a purely Maxwellian implementation using continuously varying anisotropic and inhomogeneous material parameters. An approach to the implementation of a conformal PML for higher order meshes is also presented, based on a method of normal projection for PML mesh generation around an already existing convex volume mesh of a dielectric scatterer, with automatically generated constitutive material parameters. Once the initial mesh is generated, a PML optimization method based on gradient descent is implemented to most accurately match the PML material parameters to the geometrical interface. The numerical results show that the implementation of a conformal PML in the higher order finite-element modeling paradigm dramatically reduces the reflection error when compared to traditional PMLs with piecewise constant material parameters. The ability of the new PML to accurately and efficiently model scatterers with a large variation in geometrical shape and those with complex material compositions is demonstrated in examples of a dielectric almond and a continuously inhomogeneous and anisotropic transformation-optics cloaking structure, respectively.
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- Award ID(s):
- 1646562
- PAR ID:
- 10076263
- Date Published:
- Journal Name:
- IEEE transactions on antennas and propagation
- Volume:
- 65
- Issue:
- 12
- ISSN:
- 0018-926X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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