Efficient Multigrid Methods for Semi-implicit Landau-Lifshitz Schemes in Micromagnetic Simulations

Abstract

An efficient aggregation-based multigrid approach is developed in this work to solve linear algebraic systems generated by discretizing the Landau-Lifshitz equation in micromagnetics. The discretization couples the first-order backward differentiation formula with first-order extrapolation, and the underlying equation describes magnetization dynamics within micromagnetic simulations. For these linear systems, standard iterative methods and conventional multigrid solvers intended for symmetric problems suffer from deteriorated convergence under mesh refinement and poor computational performance. Existing algorithms struggle to handle the system's intrinsic features, namely its sparse spectral properties and non-symmetric matrix structure. To overcome these limitations, a multigrid framework is constructed, where smoothing and coarse-grid correction operators are customized for the target linear system. Numerical tests confirm the robustness and mesh-independent convergence of the resulting method. Compared with well-established Krylov-subspace solvers and conventional multigrid techniques, the aggregation-based multigrid solver cuts down iteration counts and overall computational cost considerably, while producing faithful representations of magnetization dynamics.