Optimal $L^2$ convergence of renormalized Crank-Nicolson mass lumping FEMs for the Landau-Lifshitz equation

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 $L^2$ error estimates of the proposed scheme. Numerical experiments are presented to illustrate the temporal and spatial convergence of the algorithm.