Optimal L

The paper is concerned with a linearly semi-implicit renormalized Crank–Nicolson mass lumping finite element method (FEM) for solving the Landau–Lifshitz equation, which models the dynamics of magnetization under a non-convex constraint of unit magnetization length. The numerical solutions preserve the unit length at all finite element nodes by means of renormalizations. Based on a new superconvergence result and a new geometric relation between the errors of the auxiliary solution and the renormalized numerical solution, we rigorously establish optimal L2 error estimates of the proposed scheme. Numerical experiments are presented to illustrate the temporal and spatial convergence of the algorithm.