TY - JOUR
T1 - Energy Stable Time Domain Finite Element Methods for Nonlinear Models in Optics and Photonics
AU - Anees , Asad
AU - Angermann , Lutz
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 511
EP - 541
PY - 2022
DA - 2022/06
SN - 19
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/20657.html
KW - Finite element analysis, nonlinear Maxwell’s equations, energy stability, convergence analysis, error estimate, time domain analysis.
AB - <p style="text-align: justify;">Novel time domain finite element methods are proposed to numerically solve the
system of Maxwell’s equations with a cubic nonlinearity in the spatial 3D case. The effects of linear
and nonlinear electric polarization are precisely modeled in this approach. In order to achieve an
energy stable discretization at the semi-discrete and the fully discrete levels, a novel technique is
developed to handle the discrete nonlinearity, with spatial discretization either using edge and face
elements (Néd<span style="text-align: justify;">é</span>lec-Raviart-Thomas) or discontinuous spaces and edge elements (Lee-Madsen). In
particular, the proposed time discretization scheme is unconditionally stable with respect to the
electromagnetic energy and is free of any Courant-Friedrichs-Lewy-type condition. Optimal error
estimates are presented at semi-discrete and fully discrete levels for the nonlinear problem. The
methods are robust and allow for discretization of complicated geometries and nonlinearities of
spatially 3D problems that can be directly derived from the full system of nonlinear Maxwell’s
equations.</p>