Numerical analysis of relaxed micromagnetics by penalised finite elements

Some micromagnetic phenomena in rigid (ferro-)magnetic materials can be modelled by a non-convex minimisation problem. Typically, minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy. Solutions exist in a generalised sense and the observed microstructure can be described in terms of Young measures. A relaxation by convexifying the energy density resolves the essential macroscopic information. The numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations. The lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed. An a priori and a posteriori error analysis is presented for a penalised version of the side-restriction that the modulus of the magnetic field is bounded pointwise. Residual-based adaptive algorithms are proposed and experimentally shown to be efficient.