In this paper, time-domain finite element methods for the full system of Maxwell's equations with cubic nonlinearities in 3D are presented, including a selection of computational experiments. The new capabilities of these methods are to efficiently model linear and nonlinear effects of the electrical polarization. The novel strategy has been developed to bring under control the discrete nonlinearity model in space and time. It results in energy stable discretizations both at the semi-discrete and the fully discrete levels, with spatial discretization using edge and face elements (Nédeléc-Raviart-Thomas formulation). In particular, the proposed time discretization schemes are unconditionally stable with respect to a specially defined nonlinear electromagnetic energy, which is an upper bound of the electromagnetic energy commonly used. The approaches presented prove to be robust and allow the modeling of 3D optical problems that can be directly derived from the full system of Maxwell's nonlinear equations, and allow the treatment of complex nonlinearities and geometries of various physical systems coupled with electromagnetic fields.

中文翻译：

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3D 非线性麦克斯韦方程组的能量稳定时域有限元方法

在本文中，介绍了具有 3D 三次非线性的麦克斯韦方程组完整系统的时域有限元方法，包括一系列计算实验。这些方法的新功能是有效地模拟电极化的线性和非线性效应。已经开发出新颖的策略来控制空间和时间上的离散非线性模型。它在半离散和完全离散的水平上产生能量稳定的离散化，使用边缘和面元素进行空间离散化（Nédeléc-Raviart-Thomas 公式）。特别是，所提出的时间离散化方案对于专门定义的非线性电磁能是无条件稳定的，这是常用的电磁能的上限。