发表机构
Universität Duisburg-Essen（杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对时变系数和非线性E-H耦合的麦克斯韦方程，提出有限元分析，证明有限时间一致收敛和无限时间指数衰减，并给出充分条件及反例验证。
AI 中文摘要
本文分析了具有时变系数以及总电流密度和Silver-Müller型边界条件中非线性E-H耦合的麦克斯韦方程的有限元逼近。非线性耦合和时变材料参数阻碍了标准分析工具的使用，包括离散紧致性和Minty-Browder论证，使得数值分析特别具有挑战性。作为主要创新点，我们建立了两个主要贡献：在每个有限时间区间$[0,T]$上的一致收敛性和在无限时间区间$[0,\infty)$上的指数稳定性。一致收敛性通过在时变Hilbert空间框架内的Cauchy型论证建立，该框架完全绕过了Minty技巧和离散紧致性论证。此外，在非线性满足正齐次性假设下，通过由连续离散能量之比产生的非线性sup-max问题来分析指数稳定性。利用这一策略，我们最终在特定的充分条件下证明了完全离散的无条件指数衰减结果。在缺乏该条件的情况下，一个具体的反例表明能量衰减不成立。提供了数值实验以验证理论发现。
英文摘要
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 $[0,\infty)$. 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.
