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The full Landau-Lifshitz-Gilbert equation with periodic material coefficientsand natural boundary condition is employed to model the magnetization dynamicsin composite ferromagnets. In this work, we establish the convergence betweenthe homogenized solution and the original solution via a Lax equivalencetheorem kind of argument. There are a few technical difficulties, including: 1)it is proven the classic choice of corrector to homogenization cannot providethe convergence result in the $H^1$ norm; 2) a boundary layer is induced due tothe natural boundary condition; 3) the presence of stray field give rise to amultiscale potential problem. To keep the convergence rates near the boundary,we introduce the Neumann corrector with a high-order modification. Estimates onsingular integral for disturbed functions and boundary layer are deduced, toconduct consistency analysis of stray field. Furthermore, inspired by lengthconservation of magnetization, we choose proper correctors in specificgeometric space. These, together with a uniform $W^{1,6}$ estimate on originalsolution, provide the convergence rates in the $H^1$ sense.

Homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary condition