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S. Ferraz-Leite, J. Melenk, D. Praetorius:
"Finite element discretization of a reduced model in thin- lm micromagnetics";
Talk: MAFELAP 2009 - The Mathematics of Finite Elements and Applications, Uxbridge (invited); 2009-06-09 - 2009-06-12.

We consider the reduced model proposed in [DeSimone, Kohn, Müller,
Otto, Schäfer, 2001] which is consistent with the prior works
[Bryant, Suhl, 1989] and [van den Berg 1986] and is valid for
sufficiently large and thin ferromagnetic samples.

Let ω denote a bounded Lipschitz domain in #R² with diameter
l=1. This domain represents our ferromagnetic sample whose thickness
is neglected for simplicity. Here, we consider a uniaxial material
with in-plane easy axis e₁. With an applied exterior field
f:ω -> R², we seek a minimizer m^* of the reduced energy

e(m)=1/2 ||grad u||_L^2(R³)²+q/2 ||m₂²||_L^2(ω)-(f,m)_L^2(ω)

with the convex side constraint |m|<=1. Here u:R³ -> R denotes
the magnetostatic potential related to the magnetization via

(grad u, grad φ)_L^2(R³)
= (m, grad φ)_L^2(ω) for all φ in D(R³).

We stress that the convexity of the reduced minimization problem
which is highly degenerate for vanishing anisotropic energy
contribution q=0. Moreover, an additional difficulty arises due to
the non-local behavior of u which is coupled with m.

We analyze the model problem and give a precise and appropriate
functional analytic framework in a certain subspace of H^1/2
(div,ω). Existence and uniqueness of a minimizer m^* in our
functional setting is proven. We propose a numerical discretization
strategy by use of lowest-order Raviart-Thomas finite elements.
Numerical examples conclude the talk.

Thin-film micromagnetics, micromagnetics, Raviart-Thomas