Iterative methods for the saddle-point problem arising from the $H_C/E_I$ formulation of the eddy current problem.

This paper is concerned with the solution of the linear system arising from a finite element approximation of the time-harmonic eddy current problem. We consider the ${\bf H}_C/{\bf E}_I$ formulation introduced and analyzed in [A.~M. Alonso Rodr{\'\i}guez, R.~Hiptmair, and A.~Valli, {\it Numer.\ Methods Partial Differential Equations}, 21 (2005), pp.~742--763], where an optimal error estimate for the finite element approximation using edge elements of the first order is proved. Now we propose and analyze iterative procedures for the solution of the resulting linear system based on the physical decomposition of the computational domain in an insulating region and a conducting one. If the insulator does not contain any nonbounding cycle, we prove that the Dirichlet--Neumann iteration converges with a rate that is independent of the mesh size. In the case of a connected conductor with general topology we propose to use either a modified version of the Dirichlet--Neumann iteration or an Uzawa-like method. We compare the performance of both methods by solving four different test problems.