An Energy Stable Discontinuous Galerkin Time-Domain Finite Element Method in Optics and Photonics

Abstract

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.