Equivalence between two finite element methods for the eddy current problem (Q990207)

The authors consider the time-harmonic eddy current problem, an approximation of the full Maxwell equations. The domain is three dimensional and consists of a part occupied by conductors and a part occupied by dielectrics. They investigate two different finite element approximations of this model based on certain weak formulations. The first one involves the magnetic field in the conductor and a scalar potential in the dielectric. The second one uses the magnetic field in the whole domain and a Lagrange multiplier in the dielectric. The authors show that the two formulations are equivalent. The second formulation leads to a singular system of linear equations. A third formulation is also given which leads to a system with a non-singular matrix. The authors prove that this third formulation is equivalent to the other two formulations.