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Volume 19, Issue 4
Energy Stable Time Domain Finite Element Methods for Nonlinear Models in Optics and Photonics

Asad Anees & Lutz Angermann

Int. J. Numer. Anal. Mod., 19 (2022), pp. 511-541.

Published online: 2022-06

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  • Abstract

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 (Nédélec-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.

  • Keywords

Finite element analysis, nonlinear Maxwell’s equations, energy stability, convergence analysis, error estimate, time domain analysis.

  • AMS Subject Headings

35Q61, 65M60, 78A60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-511, author = {Asad and Anees and and 23778 and and Asad Anees and Lutz and Angermann and and 23779 and and Lutz Angermann}, title = {Energy Stable Time Domain Finite Element Methods for Nonlinear Models in Optics and Photonics}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {4}, pages = {511--541}, abstract = {

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 (Nédélec-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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/20657.html} }
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 -

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 (Nédélec-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.

Asad Anees & Lutz Angermann. (2022). Energy Stable Time Domain Finite Element Methods for Nonlinear Models in Optics and Photonics. International Journal of Numerical Analysis and Modeling. 19 (4). 511-541. doi:
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