Finite Element Analysis of Maxwell's Equations with Time-Varying Coefficients and Nonlinear E-H Coupling: Convergence and Exponential Decay
Abstract
This paper analyzes finite element approximations for Maxwell's equations with time-dependent coefficients and nonlinear E-H coupling in both the total current density and the Silver-Müller-type boundary condition. The nonlinear coupling and the time-varying material parameters prevent the use of standard analytical tools, including the discrete compactness property and the Minty-Browder argument, making the numerical analysis particularly challenging. As the main novelty, we establish two primary contributions: Uniform convergence on every finite time interval [0,T] and exponential stability on the infinite time horizon [0,∞). The uniform convergence is established via a Cauchy-type argument within a time-dependent Hilbert space framework that entirely bypasses Minty's trick and the discrete compactness argument. Furthermore, under a positive homogeneity assumption on the nonlinearities, the exponential stability is analyzed by means of a nonlinear sup-max problem arising from ratios of consecutive discrete energies. With this strategy, we ultimately prove a fully discrete unconditional exponential decay result under a specific sufficient condition. In the absence of this condition, a concrete counterexample shows that the energy decay fails to hold. Numerical experiments are provided to validate the theoretical findings.
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BibTeX
@misc{renner2026finite,
title = {{Finite Element Analysis of Maxwell's Equations with Time-Varying Coefficients and Nonlinear E-H Coupling: Convergence and Exponential Decay}},
author = {Jan Renner and Irwin Yousept},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32967},
eprint = {2609.32967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32967},
}
RIS
TY - UNPB
TI - Finite Element Analysis of Maxwell's Equations with Time-Varying Coefficients and Nonlinear E-H Coupling: Convergence and Exponential Decay
AU - Jan Renner
AU - Irwin Yousept
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32967
UR - https://arxiv.org/abs/2609.32967
AB - This paper analyzes finite element approximations for Maxwell's equations with time-dependent coefficients and nonlinear E-H coupling in both the total current density and the Silver-Müller-type boundary condition. The nonlinear coupling and the time-varying material parameters prevent the use of standard analytical tools, including the discrete compactness property and the Minty-Browder argument, making the numerical analysis particularly challenging. As the main novelty, we establish two primary contributions: Uniform convergence on every finite time interval [0,T] and exponential stability on the infinite time horizon [0,∞). The uniform convergence is established via a Cauchy-type argument within a time-dependent Hilbert space framework that entirely bypasses Minty's trick and the discrete compactness argument. Furthermore, under a positive homogeneity assumption on the nonlinearities, the exponential stability is analyzed by means of a nonlinear sup-max problem arising from ratios of consecutive discrete energies. With this strategy, we ultimately prove a fully discrete unconditional exponential decay result under a specific sufficient condition. In the absence of this condition, a concrete counterexample shows that the energy decay fails to hold. Numerical experiments are provided to validate the theoretical findings.
ER -