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

The aim of this paper is to analyze a numerical method for solving the 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 nonmagnetic conductors carrying currents. The topology of the domain of each material is not assumed to be trivial. The resulting variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements. Transmission conditions are imposed by means of a piecewise linear Lagrange multiplier on the surface separating the domains of both potentials. Error estimates for the numerical method are proved and the results of some numerical tests are reported to assess the performance of the method.