Convergence and stability of a numerical method for micromagnetics (Q1014240)

This paper studies convergence and stability of a numerical method for micromagnetics, which applies a nonconforming finite element method and an artificial boundary method to a multiatomic Young measure relaxation model. Through considering the convex-hull relaxation of the problem \[ \begin{aligned} &\min_{m\in A} E(m),\\ &\text{with }\quad E(m)=\int_{\Omega}\varphi(m)\,dx-\int_\Omega H\cdot m\,dx+\frac{1}{2}\int_{R^n}|\nabla u_m|^2\,dx, \quad n=2,3, \\ &\text{and }\quad {\mathcal A}=\{m: m\in (L^2(\Omega))^n \text{ and } |m|=1, \text{ a.e. }x\in\Omega\},\\ &\text{s.t.}\\ & \text{div}(-\nabla u_m+m\chi_\Omega)=0, \quad\text{in }\;H^{-1}(R^n),\\ &u_m\to 0, \quad\hbox{as }\;|x|\to \infty, \end{aligned} \] the authors show that their numerical method is stable and that the solution set of the continuous problem is well approximated by those of the discrete problems. Finally, some numerical examples are given to illustrate the convergence and stability of the numerical method. This work is carefully done and the results of this paper are interesting. The paper is novel, clear and concise.