Unconditional optimal error estimates of linearized second-order BDF Galerkin FEMs for the Landau-Lifshitz equation

In this paper, we establish the unconditionally optimal error estimates of linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) for the Landau-Lifshitz equation describing the magnetic behavior in ferromagnetic materials. By using the temporal-spatial error splitting techniques, we split the error between the exact solution and the numerical solution into two parts which are called the temporal error and the spatial error. First, we analyze the temporal error by introducing a time-discrete system and derive some regularity of the solution of the time-discrete system. Second, by the above achievements, we obtain the τ-independent spatial error and the boundedness of the numerical solution in L^∞-norm. Then, the optimal L² and H¹ error estimates for r-th order FEMs (r=1,2) are derived without any restriction on the time step size. Numerical results in both two and three dimensional spaces are presented to confirm the theoretical predictions and demonstrate the efficiency of the methods.