Numerical analysis of a model for ferromagnetic shape memory thin films

We give a model for the deformation and magnetization of a single crystal ferromagnetic shape memory thin film under the influence of an applied magnetic field. The energy is nonconvex since it models multiple phases and symmetry-related variants of the crystal structure. Nonconvexity is also presented by the magnetic saturation condition which requires the magnetization to have a constant magnitude. We propose a class of finite element methods and prove a rate of convergence for the minimum thin film energy. In addition to the challenge of analyzing a nonconvex energy, the analysis overcomes the challenge presented by contributions to the energy that are naturally defined in the reference configuration for the elastic energy and in the spatial frame for the magnetic energy. We present numerical computations for the deformation and magnetization of a Ni₂MnGa thin film that exhibit the convergence rate given by analysis.