Fourier-spectral method for the Landau–Lifshitz–Gilbert equation in micromagnetism

The Landau–Lifshitz–Gilbert (LLG) equation models the temporal evolution of magnetization in continuum ferromagnets. The LLG equation has a nonconvex constraint and is highly nonlinear. In this paper, we will use the Fourier-spectral method for approximating the solution of the LLG equation with the nonconvex constraint. We consider the penalty problem and show the stability and convergence of the approximate penalty problem, and then we show the convergence of the penalty problem to a (weak) solution of the LLG equation. Computational experiments and comparison with other numerical methods are presented to demonstrate the effectiveness of the proposed method.