Error Estimates of Projection Finite Element Method for Landau-Lifshitz Equation

This paper establishes a Crank-Nicolson projection finite element scheme for the Landau-Lifshitz equation, and derives unconditional optimal error estimates. By employing the temporal-spatial error splitting technique, the error between the exact solution and the numerical solution is decomposed into three components: the temporal error, the spatial error and the spherical projection error. First, the temporal error is analyzed by introducing a linearized time-discrete system, and the regularity of the solution for this time-discrete system is rigorously proved. Second, based on the time-discrete system, a spatial error equation independent of the time step τ is derived. Under the condition that the time step and spatial step are unrestricted, the optimal error estimate of the projection finite element method in the L² norm is demonstrated. Moreover, numerical experiments on model problems are conducted to validate the theoretical results and the effectiveness of the proposed method.