In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell's equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with $Q_k$-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy. The approaches presented prove to be robust and allow the modeling of optical problems and the treatment of complex nonlinearities as well as geometries of various physical systems coupled with electromagnetic fields.

翻译：本文研究了用于光学与光子学中完整麦克斯韦方程组时域间断伽辽金有限元方法，包括半离散误差估计的完整证明。该类方法的新能力在于能高效模拟线性和非线性效应，例如克尔非线性效应。本文提出了半离散和全离散两个层面上的能量稳定离散格式。特别地，所提出的半离散格式在采用$Q_k$型单元的笛卡尔网格上对空间变量具有最优收敛性，而全离散格式相对于特殊定义的非线性电磁能具有条件稳定性。该方法被证明具有鲁棒性，能够模拟光学问题，处理复杂非线性以及耦合约电磁场的各类物理系统的几何结构。