Variational methods for solving numerically magnetostatic systems

In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of T-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the weighted regularization method.