Analysis of a backward Euler-type scheme for Maxwell’s equations in a Havriliak–Negami dispersive medium

¹ 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: lilian@ntu.edu.sg

Received: 15 July 2020
Accepted: 21 January 2021

Abstract

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.