For the Maxwell’s equations in a Havriliak–Negami (H-N) dispersive medium, the associated energy dissipation law has not been settled at both continuous level and discrete level. In this paper, we rigorously show that the energy of the H-N model can be bounded by the initial energy and the model is well-posed. We analyse a backward Euler-type semi-discrete scheme, and prove that the modified discrete energy decays monotonically in time. Such a strong stability ensures that the scheme is unconditionally stable. We also introduce a fast temporal convolution algorithm to alleviate the burden of the history dependence in the polarisation relation involving the singular kernel with the Mittag-Leffler function with three parameters. We provide ample numerical results to demonstrate the efficiency and accuracy of a full-discrete scheme via a spectra-Galerkin method in two dimensions. Finally, we consider an interesting application in the recovery of complex relative permittivity and some related physical quantities.
Keywords: Maxwell’s equations, Havriliak–Negami dispersive medium, strong stability, unconditionally stable scheme, fast temporal convolution algorithm
@article{M2AN_2021__55_2_479_0,
author = {Yang, Yubo and Wang, Li-Lian and Zeng, Fanhai},
title = {Analysis of a backward {Euler-type} scheme for {Maxwell{\textquoteright}s} equations in a {Havriliak{\textendash}Negami} dispersive medium},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {479--506},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {2},
doi = {10.1051/m2an/2021004},
mrnumber = {4229195},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021004/}
}
TY - JOUR AU - Yang, Yubo AU - Wang, Li-Lian AU - Zeng, Fanhai TI - Analysis of a backward Euler-type scheme for Maxwell’s equations in a Havriliak–Negami dispersive medium JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 479 EP - 506 VL - 55 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021004/ DO - 10.1051/m2an/2021004 LA - en ID - M2AN_2021__55_2_479_0 ER -
%0 Journal Article %A Yang, Yubo %A Wang, Li-Lian %A Zeng, Fanhai %T Analysis of a backward Euler-type scheme for Maxwell’s equations in a Havriliak–Negami dispersive medium %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 479-506 %V 55 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021004/ %R 10.1051/m2an/2021004 %G en %F M2AN_2021__55_2_479_0
Yang, Yubo; Wang, Li-Lian; Zeng, Fanhai. Analysis of a backward Euler-type scheme for Maxwell’s equations in a Havriliak–Negami dispersive medium. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 479-506. doi: 10.1051/m2an/2021004
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