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F. Bruckner, M. Feischl, T. Führer, P. Goldenits, M. Page, D. Praetorius, M. Ruggeri, D. Süss:
"Multiscale modeling in micromagnetics: Existence of solutions and numerical integration";
Mathematical Models & Methods in Applied Sciences, 24 (2014), 13; 2627 - 2662.

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 H1-convergence -at least for a subsequence- towards a weak solution

Micromagnetism, Landau-Lifshitz-Gilbert equation, multiscale modeling, FEM-BEM coupling, convergence analysis

http://dx.doi.org/10.1142/S0218202514500328