Convergent geometric integrator for the Landau‐Lifshitz‐Gilbert equation in micromagnetics

Abstract We consider a lowest‐order finite element scheme for the Landau‐Lifshitz‐Gilbert equation (LLG) which describes the dynamics of micromagnetism. In contrast to previous works, we examine LLG with a total magnetic field which is induced by several physical phenomena described in terms of exchange energy, anisotropy energy, magnetostatic energy, and Zeeman energy. In our numerical scheme, the highest‐order term which stems from the exchange energy, is treated implicitly, whereas the remaining energy contributions are computed explicitly. Therefore, only one sparse linear system has to be solved per time‐step. The proposed scheme is unconditionally convergent to a global weak solution of LLG. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)