Stabilization methods in relaxed micromagnetism

Funken, Stefan A.; Prohl, Andreas

ESAIM: 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 ].

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