Numerical Analysis for a Macroscopic Model in Micromagnetics

The macroscopic behavior of stationary micromagnetic phenomena can be modeled by a relaxed version of the Landau--Lifshitz minimization problem. In the limit of large and soft magnets $\Omega$, it is reasonable to exclude the exchange energy and convexify the remaining energy densities. The numerical analysis of the resulting minimization problem, \begin{align*} \min E_0^{**}({\bf m})\text{ amongst }{\bf m}:\Omega\to\mathbb{R}^d\text{ with } |{\bf m}(x)|\le1\text{ for almost every }x\in\Omega, \end{align*} for $d=2,3$, faces difficulties caused by the pointwise side-constraint $|{\bf m}|\le1$ and an integral over the whole space $\mathbb{R}^d$ 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 $\mathcal{P}$ acting on functions exclusively defined on $\Omega$. The discretization with piecewise constant discrete magnetizations leads to edge-oriented boundary integrals, the implementation of which and related numerical quadrature are discussed, as are 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 performance of the uniform and adaptive mesh-refining algorithms.