Paper Info
Title
Numerical Analysis for a Macroscopic Model in Micromagnetics
Abstract
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.
Year
DOI
Venue
2005
10.1137/S003614290343565X
SIAM J. Numerical Analysis
Keywords
Field
DocType
numerical analysis,macroscopic model,adaptive mesh-refining algorithm,stray field energy,numerical quadrature,numerical experiment,exchange energy,lifshitz minimization problem,adaptive algorithm,remaining energy density,bf m,micromagnetics,relaxation,penalty method,microstructure,convergence rate,magnetization,maxwell equation,newton potential
Newtonian potential,Discretization,Mathematical optimization,Mathematical analysis,Numerical integration,Rate of convergence,Integral element,Numerical analysis,Piecewise,Mathematics,Pointwise
Journal
Volume
Issue
ISSN
42
6
0036-1429
Citations 
PageRank 
References 
5
0.73
2
Authors
2
Name
Order
Citations
PageRank
C Carstensen1944163.02
Dirk Praetorius212122.50