Error analysis of numerical scheme for 3D Maxwell-Landau-Lifshitz system IvanCimr´ak * November 25, 2005 Keywords Micromagnetism, Maxwell equations, numerical scheme, ferromag- nets, error analysis Subject classification 35K55 Abstract We suggest a linear numerical scheme solving strongly nonlinear cou- pled Maxwell-Landau-Lifshitz (Maxwell-LL) system describing ferromagnetic phenomena. Using recent results on the regularity of the solutions to the Maxwell-LL system we are able to prove convergence and to derive the er- ror estimates for this numerical scheme. We provide numerical examples which confirm the theoretical results. 1 Introduction For the electromagnetic applications requiring small scale analysis the Landau- Lifshitz (LL) model is widely used. This model describes ferromagnetic phe- nomena in the scales ranging from nanometers to micrometers. The equation governing the LL model reads as ∂ t m = −α 1 m × (Δm + H) − αm × (m × (Δm + H)), in Ω ⊂ R 3 , (1) where constants α 1 ,α are of physical origin linked to the gyromagnetic factor, and the symbols m, H denote magnetization and magnetic field, respectively. For more complex model one has to incorporate Maxwell’s equations. We con- sider the quasi-static case governed by equation ∂ t H + β 1 ∇×∇× H = −β∂ t m, (2) The constants satisfy α> 0,β ≥ 0. We are interested in the case when the solutions are space-periodic functions. This model arises from various physical and engineering applications such as electromagnetic wave propagation or antennas. It can be used also for the study of electromagnetic behavior of the materials with a periodical structure. The function spaces that correspond to space-periodic functions will be de- noted by a subscript “per”, e.g. W 1,2 per (Ω). * The work of Ivan Cimr´ak was supported by the IUAP/V-P5/34 project (at Ghent Uni- versity) of the DWTC of the Belgian Federal Government. 1