Stabilization methods in relaxed micromagnetism
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 5, pp. 995-1017.

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potential $u$ and magnetization $𝐦$. In [C. Carstensen and A. Prohl, Numer. Math. 90 (2001) 65-99], the conforming $P1-{\left(P0\right)}^{d}$-element in $d=2,3$ spatial dimensions is shown to lead to an ill-posed discrete problem in relaxed micromagnetism, and suboptimal convergence. This observation motivated a non-conforming finite element method which leads to a well-posed discrete problem, with solutions converging at optimal rate. In this work, we provide both an a priori and a posteriori error analysis for two stabilized conforming methods which account for inter-element jumps of the piecewise 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 ].

DOI : https://doi.org/10.1051/m2an:2005043
Classification : 65K10,  65N15,  65N30,  65N50,  73C50,  73S10
Mots clés : micromagnetics, stationary, nonstationary, microstructure, relaxation, nonconvex minimization, degenerate convexity, finite elements methods, stabilization, penalization, a priori error estimates, a posteriori error estimates
@article{M2AN_2005__39_5_995_0,
author = {Funken, Stefan A. and Prohl, Andreas},
title = {Stabilization methods in relaxed micromagnetism},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {995--1017},
publisher = {EDP-Sciences},
volume = {39},
number = {5},
year = {2005},
doi = {10.1051/m2an:2005043},
zbl = {1079.78031},
mrnumber = {2178570},
language = {en},
url = {www.numdam.org/item/M2AN_2005__39_5_995_0/}
}
Funken, Stefan A.; Prohl, Andreas. Stabilization methods in relaxed micromagnetism. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 5, pp. 995-1017. doi : 10.1051/m2an:2005043. http://www.numdam.org/item/M2AN_2005__39_5_995_0/

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