A unified FEM–BEM approach for electro–magnetostatics and time-harmonic eddy-current problems

In this paper we consider a unified approach for solving the electro-magnetostatics problem and the eddy current problem in terms of suitable potentials. A new variational formulation is proposed, in which standard results of potential theory are used to reduce the problem in the external domain to an integral equation on the boundary of the conductor. The existence and uniqueness of the solution is proved, by showing that the associated sesquilinear form is coercive. A numerical approximation scheme, based on nodal finite elements in the conductor and boundary elements on its boundary, is devised and proved to be convergent. It is also shown that the solution of the time-harmonic eddy current problem tends to the solution of the electro-magnetostatics problem as the frequency tends to 0. The same convergence holds, uniformly with respect to the mesh size, for the finite element solutions.