BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: Unconditional weak convergence to weak solutions

We consider the Landau–Lifshitz–Gilbert equation (LLG) that models time-dependent micromagnetic phenomena. We propose a full discretization that employs first-order finite elements in space and a second-order backward differentiation formula (BDF2) two-step method in time. In each time step, only one linear system of equations has to be solved. We employ linear interpolation in time to reconstruct the discrete space-time magnetization. We prove that the integrator is unconditionally stable and thus guarantees that a subsequence of the reconstructed magnetization converges weakly in H 1 H^{1} towards a weak solution of LLG in the space-time domain. Numerical experiments verify that the proposed integrator is indeed first-order in space and second-order in time.