In this paper we construct upper bounds for families offunctionals of the form

$$E_\varepsilon(\phi):=\int_\Omega\Big(\varepsilon |\nabla\phi|^2+\frac{1}{\varepsilon }W(\phi)\Big){\rm d}x+\frac{1}{\varepsilon }\int_{{\mathbb{R}}^N}|\nabla \bar H_{F(\phi)}|^2{\rm d}x$$

where Δ $\bar H_u$ = div { $\chi_\Omega$ u}. Particular cases of such functionals arise inMicromagnetics. We also use our technique to construct upper boundsfor functionals that appear in a variational formulation ofthe method of vanishing viscosity for conservation laws.

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization insidethe magnetic sample is of particular relevance. We here investigate mathematicallythis problem by considering the full partial differential model of Landau-Lifschitzequations triggered by a uniform (in space) external magnetic field.

The magnetization of a ferromagnetic sample solves anon-convex variational problem, where its relaxation by convexifyingthe energy density resolves relevantmacroscopic information. The numerical analysis of the relaxed modelhas to deal with a constrained convexbut degenerated, nonlocal energy functional in mixed formulation formagnetic potential u and magnetization m.In [C. Carstensen and A. Prohl, Numer. Math.90(2001) 65–99], the conforming P1 - (P0)d -element in d=2,3 spatialdimensions is shown to lead toan ill-posed discrete problem in relaxed micromagnetism, and suboptimalconvergence.This observation motivated anon-conforming finite element method which leads toa well-posed discrete problem, with solutions converging atoptimal rate.In this work, we provide both an a priori and a posteriori error analysis for twostabilized conforming methods which account for inter-element jumps of thepiecewise constant magnetization.Both methods converge at optimal rate;the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $$ \mathcal{E}_{\varepsilon}(m)=\int_{\Omega} \phi\left(x,\frac{x}{\varepsilon},m(x)\right)\,{\rm d}x-\int_{\Omega}h_e(x)\cdot m(x)\,{\rm d}x+\frac{1}{2}\int_{\mathbb R^3}|\nabla u(x)|^2\,{\rm d}x$$ of a large ferromagnetic body is obtained.

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.


We study a two-dimensional model for micromagnetics, which consists in an energy functional over S 2-valued vector fields. Bounded-energy configurations tend to be planar, except in small regions which can be described as vortices (Bloch lines in physics). As the characteristic “exchange-length” tends to 0, they converge to planar divergence-free unit norm vector fields which jump along line singularities. We derive lower bounds for the energy, which are explicit functions of the jumps of the limit. These lower bounds are proved to be optimal and are achieved by one-dimensional profiles, corresponding to Néel walls, if the jump is small enough (less than π/2 in angle), and by two-dimensional profiles, corresponding to cross-tie walls, if the jump is bigger. Thus, it provides an example of a vector-valued phase-transition type problem with an explicit non-one-dimensional energy-minimizing transition layer. We also establish other lower bounds and compactness properties on different quantities which provide a good notion of convergence and cost of vortices.

Γ-convergence techniques and relaxation results of constrained energy functionals are used to identify the limiting energy as the thickness ε approaches zero of a ferromagnetic thin structure $\Omega_\varepsilon=\omega\times(-\varepsilon,\varepsilon)$, $\omega\subset\mathbb R^2$, whose energy is given by $$ {\cal E}_{\varepsilon}({\overline{m}})=\frac{1}{\varepsilon} \int_{\Omega_{\varepsilon}}\left(W({\overline{m}},\nabla{\overline{m}}) +{\frac{1}{2}}\nabla {\overline{u}}\cdot {\overline{m}}\right)\,{\rm d}x $$ subject to $$ \hbox{div}(-\nabla {\overline{u}}+{\overline{m}}\chi_{\Omega_\varepsilon})=0 \quad\hbox{ on }\mathbb R^3, $$ and to the constraint $$ |\overline{m}|=1 \hbox{ on }\Omega_\varepsilon, $$ where W is any continuous function satisfying p-growth assumptions with p> 1. Partial results are also obtained in the case p=1, under an additional assumption on W.

This paper is devoted to the practical computation of the magnetic potential induced by a distribution of magnetization in the theory of micromagnetics. The problem turns out to be a coupling of an interior and an exterior problem. The aim of this work is to describe a complete method that mixes the approaches of Ying [12] and Goldstein [6] which consists in constructing a mesh for the exterior domain composed of homothetic layers. It has the advantage of being well suited for catching the decay of the solution at infinity and giving a rigidity matrix that can be very efficiently stored. All aspects are described here, from the practical construction of the mesh, the storage of the matrix, the error estimation of the method, the boundary conditions and a simple preconditionning technique. At the end of the paper, a typical computation of a uniformly magnetized ball is done and compared to the analytic solution. This method gives a natural alternatives to boundary elements methods for 3D computations.