Efficient Numerics for Thin Film Micromagnetism

We consider numerical methods for a reduced 2D-model of micromagnetism by A. DeSimone, R.V. Kohn, S. Mueller and F. Otto that describes the response of a soft ferromagnetic thin film to an applied magnetic field. The micromagnetic energy is given as a non-local functional of magnetizations m which are defined as 2D-fields of unit length on the cross section of the film.

Our discretization is based on a conformal Galerkin-Ansatz using Raviart-Thomas elements. A two-step algorithm is presented that solves the minimization problem for the energy numerically. Step 1 consists of an Interior point method for a convex variational problem, and in step 2 a minimizer satisfying the original non-convex constraint |m|=1 is generated as the solution of a Hamilton-Jacobi equation. H-matrices are used in step 1 to deal with non-sparse Hessians due to the non-locality of the energy.