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 BDF2-type 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 $H^1$ 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.

💡 Research Summary

The paper addresses the numerical solution of the Landau‑Lifshitz‑Gilbert (LLG) equation, a nonlinear, constrained PDE governing the dynamics of magnetization in ferromagnetic bodies. The authors propose a fully discrete scheme that combines first‑order conforming finite elements in space with a two‑step backward differentiation formula (BDF2) in time. The key novelty lies in the construction of a node‑wise discrete tangent space that enforces the pointwise orthogonality condition $m\cdot\partial_t m=0$ at every mesh vertex, while the unit‑length constraint $|m|=1$ is imposed directly on the nodal values.

Algorithmic structure:

1. An initial velocity $v_0^h$ is computed by solving a linear variational problem on the tangent space of the initial magnetization $m_0^h$.
2. A first‑order predictor $m_1^h = m_0^h + \tau v_0^h$ supplies the second BDF2 datum.
3. For each subsequent time step $j=1,\dots,N-1$, a predicted magnetization $c m_{j+1}^h = 2 m_j^h - m_{j-1}^h$ is formed, then a linear system on the tangent space of $c m_{j+1}^h$ yields the velocity $v_j^h$, and finally a corrected magnetization $m_{j+1}^h = \frac{4}{3}m_j^h - \frac{1}{3}m_{j-1}^h + \frac{2}{3}\tau v_j^h$ is updated. Each step requires solving only one linear system, in contrast to earlier implicit midpoint schemes that need a nonlinear solve.

Stability and convergence:
Using the Lax‑Milgram lemma, the linear systems are shown to be uniquely solvable. Energy estimates derived from the discrete variational formulation provide uniform $H^1$‑bounds for the reconstructed space‑time magnetization $m_{h\tau}$ and its forward/backward interpolants. By Banach‑Alaoglu and Rellich‑Kondrachov compactness arguments, a subsequence converges weakly in $H^1(\Omega_T)$ (and strongly in $L^2(\Omega_T)$) to a limit $m$. The limit satisfies the weak formulation of LLG (properties (i)–(iii) in the classical definition) and, under a mild CFL‑type condition $\tau=o(h^2)$ required only for the first and last steps, also fulfills the energy inequality (property (iv)). This yields unconditional weak convergence: the scheme converges without any restriction on the ratio $\tau/h$, apart from the technical condition needed for the energy inequality.

Relation to prior work:
Earlier weak‑convergence results for LLG include the implicit midpoint method