Time domain finite element method for linear and nonlinear models in electromagnetics and optics

Abstract

In this thesis, we develop a time domain nite element method for linear and nonlinear models in Electromagnetics and Optics. In the linear case, a weak formulation is derived for the electric and magnetic elds with ap- propriate initial and boundary conditions, and the problem is discretized both in space and time. Nedelec curl-conforming and Raviart-Thomas div- conforming nite elements are used to discretize in space the electric and magnetic elds, respectively. The backward Euler and symplectic schemes are applied to discretize the linear problem in time. For this linear system, we give a complete stability and error analysis. In addition, computational experiments are presented to validate the method; the electric and magnetic elds are visualized. The method also allows to treat complex geometries of various physical systems coupled to electromagnetic elds in 3D. In the next part of thesis, we extend the linear nite element method to time domain nite element methods for the full system of Maxwell's equa- tions with cubic nonlinearities in 3D. For the rst time, stability and error estimates are presented for this type of problem. The new capabilities of these methods are to eciently model linear and nonlinear eects of the electrical polarization. The novel strategy has been developed to bring un- der 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 either using discontinuous spaces and ed- ge elements (Lee-Madsen formulation) or edge and face elements (Nedelec- Raviart-Thomas formulation). To verify the stability, a novel \nonlinear electromagnetic energy is introduced, which is stronger than the the com- monly used (linear) electromagnetic energy. It turns out that the propo- sed time discretization scheme is unconditionally stable with respect to this energy. The presented computational experiments demonstrate that the pro- posed approaches prove to be robust and allow the modeling of 3D optical problems that can be directly derived from the full system of Maxwell's non- linear equations, and also allow the treatment of complex nonlinearities and geometries of various physical systems. In the last section of thesis, the time domain discretization for the nonli- near problem is extended to a discontinuous Galerkin nite element method in 2D. The energy of nonlinear Maxwell's equations at the continuous and discrete levels is described, and an error estimate at the semi-discrete level is demonstrated for the discontinuous Galerkin method.%%%%In dieser Arbeit entwickeln wir eine Finite-Elemente-Methode im Zeitbereich fur lineare und nichtlineare Modelle in Elektromagnetismus und Optik. Im linearen Fall wird eine schwache Formulierung fur das elektrische und das magnetische Feld mit geeigneten Anfangs- und Randbedingungen abgeleitet und das Problem wird sowohl raumlich als auch zeitlich diskretisiert. Nedelec Curl-konforme und Raviart-Thomas…