Modelling and mathematical results arising from ferromagnetic problems

In this article, we investigate the equations of magnetostatics for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law takes the form $$B = \mu _0 \mu _r \left( {\left| H \right|} \right)H,$$ i.e., the magnetizing field H and the magnetic induction B are collinear, but the relative permeability µr is allowed to depend on the modulus of H. We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions, and the convergence of several iterative methods, both for the original problem set in the Beppo-Levi space W 1(ℝ3), and for a finite-dimensional approximation. The theoretical results are illustrated by numerical examples, which capture the known physical phenomena.