99-18   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## On Stabilized Finite Element Methods in Relaxed Micromagnetism

The magnetization state of a ferromagnetic body is given as the solution of a non-convex variational problem. A relaxation of this model by convexifying the energy density resolves essential macroscopic information that applied physicists and engineers are after. The numerical analysis of the relaxed model faces a nonlinearly constrained convex but degenerated energy functional in mixed formulation for magnetic potential $u$ and magnetization ${\bf m}$. In \cite{CPr}, 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 behavior, even if the original problem is well-posed. This observation motivated to introduce a non-conforming method to the problem that implies a well-posed discrete problem with solutions converging at optimal rate. In this work, we introduce and discuss two stabilized conforming methods that employ inter-element jumps of the magnetization in normal direction and total magnetization jumps, respectively. Both stabilized schemes converge at optimal rates, whereas only the latter one leads to a discrete problem with unique solution.

Mathematics Subject Classification (1991): 65K10, 65N30, 73C50, 73S10, 65N15, 65N30

Keywords: micromagnetics, microstructure, relaxation, nonconvex minimization, degenerate problems, finite elements methods, stabilization, penalization, a~priori error estimates