Analysis of the Finite Element Approximation of Microstructure in Micromagnetics

Abstract

The solution to nonconvex variational problems is often characterized by microstructure. The variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite. It is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined, but that nonlinear functions of the approximate magnetization fields converge weakly (or equivalently, all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined, which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains). A norm is given to measure the convergence of this microstructure, and we prove a rate of convergence in this norm.