Abstract
We analyze the reduced model for thin-film devices in stationary micromagnetics proposed in DeSimone et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2983–2991, 2001). We introduce an appropriate functional analytic framework and prove well-posedness of the model in that setting. The scheme for the numerical approximation of solutions consists of two ingredients: The energy space is discretized in a conforming way using Raviart–Thomas finite elements; the non-linear but convex side constraint is treated with a penalty method. This strategy yields a convergent sequence of approximations as discretization and penalty parameter vanish. The proof generalizes to a large class of minimization problems and is of interest beyond the scope of thin-film micromagnetics.
Highlights
Introduction and abstract settingLet ⊆ Rn be a domain and H ⊆ L2( ) denote a continuously embedded Hilbert space with norm u 2 L2( ) u 2 H = (u, u)H
The new approach applies to a large class of problems, and we are confident that it is of interest beyond the specific application in thin-film micromagnetics
In the first part of the paper we presented a general convergence result for penalty methods
Summary
Let ⊆ Rn be a domain and H ⊆ L2( ) denote a continuously embedded Hilbert space with norm u. 3. For discrete minimization problems with finite dimensional energy space arising, for example, in the context of mathematical finance, penalty methods are well understood and convergence is established, see [21] and the references therein. Since it is known that the system of equations of the penalized discrete problem becomes ill-conditioned as ε → 0, it is natural to ask for other, more robust methods for specific applications All of these methods, are based on properties of the corresponding KKT-system or on orthogonal projections onto the admissible set. The new approach applies to a large class of problems, and we are confident that it is of interest beyond the specific application in thin-film micromagnetics It avoids use of KKT equations; it avoids estimates that require information about the Lagrange multipliers
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