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 m. In CPr], the conforming P 1 ? (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.