Numerical Analysis for a Macroscopic Model in Micromagnetics

The macroscopic behaviour of stationary micromagnetic phenomena can be modelled by a relaxed version of the Landau-Lifshitz minimization problem. In the limit of large and soft magnets Ω, it is reasonable to exclude the exchange energy and convexify the remaining energy densities. The numerical analysis of the resulting minimization problem min E 0 (m) amongst m : Ω → R with |m(x)| ≤ 1 for a.e. x ∈ Ω, for d = 2, 3, faces difficulties caused by the pointwise side-constraint |m| ≤ 1 and an integral over the whole space R for the stray field energy. This paper involves a penalty method to model the side-constraint and reformulates the exterior Maxwell equation via a nonlocal integral operator P acting on functions exclusively defined on Ω. The discretization with piecewise constant discrete magnetizations leads to edge-oriented boundary integrals. The implementation of which and related numerical quadrature is discussed as well as adaptive algorithms for automatic mesh-refinement. A priori and a posteriori error estimates provide a thorough rigorous error control of certain quantities. Three classes of numerical experiments study the penalization, empirical convergence rates, and the performance of the uniform and adaptive mesh-refining algorithms.