Multiscale Modeling in Micromagnetics: Well-Posedness and Numerical Integration

Various applications ranging from spintronic devices, giant magnetoresistance (GMR) sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau-Lifshitz-Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but therefore cannot describe the short-range exchange interaction accurately. A combination of both methods allows to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a nonlinear material law using Maxwell’s equations. In our work, we prove that under certain assumptions on the nonlinear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak H-convergence —at least for a subsequence— towards a weak solution.