Effective simulation of a macroscopic model for stationary micromagnetics

The effective behaviour of stationary micromagnetic phenomena is modelled by a convexified Landau-Lifshitz minimization problem for the limit of large and soft magnets Ω without the exchange energy. The numerical simulation of the resulting minimization problem has to overcome difficulties caused by the pointwise side-restriction ∣m∣ ⩽ 1 and the stray field energy on the unbounded domain Rd. A penalty method models the side-restriction and the exterior Maxwell equation is recast via a nonlocal integral operator P. As shown in [Numer. Anal. Macrosc. Model Micromagnet, submitted for publication; Analysis, Numerik und Simulation eines Relaxierten Modellproblems zum Mikromagnetismus, doctoral thesis, Vienna University of Technology, 2003], the discretization leads to a nonlocal problem with piecewise constant ansatz and test functions and (dense) stiffness matrices with closed form formula for their entries. This paper addresses the numerical solution with Newton-Raphson schemes and the scientific computation of effective micromagnetic simulations.