Stabilisation yields strong convergence of macroscopic magnetisation vectors for micromagnetics without exchange energy

Abstract

The convexified Landau–Lifshitz minimisation problem in micromagnetics leads to a degenerate variational problem. Therefore strong convergence of finite element approximations cannot be expected in general. This paper introduces a stabilized finite element discretization which allows for surprising the strong convergence of the discrete magnetisation fields with reduced convergence order for a uniaxial model problem. This yields a convergent method for the approximation of the Young measure which characterises the enforced microstructure for the generalized solution of the non-relaxed Landau–Lifshitz problem.