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 ].