Non-conforming finite element and artificial boundary in multi-atomic Young measure approximation for micromagnetics

In this paper, a non-conforming finite element method coupled with an artificial boundary technique is developed in a multi-atomic Young measure approximation to solve the two-dimensional variational problem for the magnetization field in micromagnetics, which has an anisotropic potential energy and a nonconvex constraint and thus can develop microstructures. Compared with the conforming finite element approach, which turns out to be unstable in the sense that spurious numerical oscillations can occur in the discrete macroscopic magnetization field, the stability and convergence of the non-conforming finite element method can be established. It is also proved that, for the uniaxial energy density, two-atomic young measure is sufficient to approximate the macroscopic magnetization field. The efficiency of the method is illustrated by some numerical examples.