Volume 55, Number 2, March-April 2021
|Page(s)||479 - 506|
|Published online||15 March 2021|
Analysis of a backward Euler-type scheme for Maxwell’s equations in a Havriliak–Negami dispersive medium
Nanhu College, Jiaxing University, 314001 Jiaxing, Zhejiang, P.R. China
2 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
3 School of Mathematics, Shandong University, 250100 Jinan, Shandong, P.R. China
* Corresponding author: email@example.com
Accepted: 21 January 2021
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.
Mathematics Subject Classification: 65N35 / 65E05 / 65N12 / 41A10 / 41A25 / 41A30 / 41A58
Key words: Maxwell’s equations / Havriliak–Negami dispersive medium / strong stability / unconditionally stable scheme / fast temporal convolution algorithm
© EDP Sciences, SMAI 2021
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