A Finite Element Method for a Nonlinear Magnetostatic Problem in Terms of Scalar Potentials

The aim of this paper is to perform the analysis of a numerical method based on scalar potentials for solving a nonlinear magnetostatic problem in a three-dimensional bounded domain containing prescribed currents and magnetic materials. The method discretizes a well-known formulation of this problem based on two scalar potentials: the total potential, defined in magnetic materials, and the reduced potential, defined in dielectric media and in non-magnetic conductors carrying currents. The topology of the magnetic materials is not assumed to be trivial, which leads to a multivalued potential. The resulting nonlinear variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements. A convergence result without regularity assumptions on the solutions is proved in both the linear and nonlinear cases. Moreover, optimal error estimates are proved for smooth functions. Numerical results for an analytical test are reported to assess the performance of the method in the case of the continuous solution being smooth.