Convergence result for the constraint preserving mid-point scheme for micromagnetism

An important progress was recently done in numerical approximation of weak solutions to a micromagnetic model equation. The problem with the nonconvex side-constraint of preserving the length of the magnetization was tackled by using reduced integration. Several schemes were proposed and their convergence to weak solutions was proved. All schemes were derived from the Landau–Lifshitz–Gilbert form of the micromagnetic equation. However, when the precessional term in the original Landau–Lifshitz (LL) form of the micromagnetic equation tends to zero, the above schemes become unusable.We propose a scheme derived from the mid-point rule for the LL form of the micromagnetic equation combined with the reduced integration. We show convergence to a weak solution of the LL equation and demonstrate the usefulness of the proposed scheme to study the limit process when the precessional parameter of the micromagnetic equation goes to zero.