AbstractsMathematics

The Landau–Lifshitz–Gilbert (LLG) equation is generally accepted as an appropriate model of phenomena observed in conventional ferromagnetic materials such as ferromagnetic thin films. In order to model the electromagnetic behaviour of the ferromagnetic material, the basic Maxwell system must be augmented by the LLG equation. The complex quantities appearing in these models are nonlocal character for some quantities, nonconvex, side-constraints and strongly nonlinear terms. These effects also cause interesting numerical approximations. An important problem in the theory of ferromagnetism is to describe noise-induced transitions between different equilibrium states. Therefore, the LLG equation needs to be modified in order to incorporate random fluctuations into the dynamics of magnetisation. The influence of noise requires a proper study of the stochastic version of the LLG equation and the Maxwell system. In this dissertation, firstly we propose a θ-linear finite element scheme for the numerical solution of the quasi-static Maxwell–Landau–Lifshitz–Gilbert (MLLG) system. Despite the strong non-linearity of the LLG equation, the proposed method results in a linear system at each time step. We prove that the finite element solutions converge to a weak solution of the MLLG system with no condition imposed on time step and space step as θ ∈ ( 1 2 , 1]. Secondly, we solve the stochastic LLG equation by using the θ-linear scheme mentioned above. We first reformulate the equation into an equation the unknown of which is differentiable with respect to the time variable. Thus the θ-linear scheme can be applied. As a consequence, we show the convergence of the finite element solutions to a weak martingale solution of the stochastic LLG equation. Finally, we consider the stochastic LLG equation coupled to the Maxwell equation, so called the stochastic MLLG system. We first apply the change of variable technique into the stochastic LLG equation, and then use the θ-linear scheme to approximate the numerical solutions of the reformulated system. We prove the convergence of the numerical solutions and the existence of weak martingale solutions of the stochastic MLLG system.