Optimal error analysis of the normalized tangent plane FEM for Landau–Lifshitz–Gilbert equation

The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{\textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.