Stabilization methods in relaxed micromagnetism
Department of Numerical Analysis, University of Ulm, 89069 Ulm, Germany. email@example.com
2 Department of Mathematics, ETHZ, 8092 Zürich, Switzerland.
Revised: 4 May 2005
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 - (P0)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 ].
Mathematics Subject Classification: 65K10 / 65N15 / 65N30 / 65N50 / 73C50 / 73S10
Key words: Micromagnetics / stationary / nonstationary / microstructure / relaxation / nonconvex minimization / degenerate convexity / finite elements methods / stabilization / penalization / a priori error estimates / a posteriori error estimates.
© EDP Sciences, SMAI, 2005