  Various applications ranging from spintronic devices, giant magnetoresistance
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 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 non-linear material law using the Maxwell
equations. In our work, we prove that under certain assumptions on the
non-linear 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 $H^1$-convergence (at least for
a subsequence) towards a weak solution.
