An iterative Domain Decomposition Method (DDM) is applied into a mixed formulation of magnetic field problems like eddy current and magnetostatic ones.From the engineering point of view, these magnetic field problems are often formulated by neglecting any gauge conditions, where the magnetic vector potential is only one unknown function. The formulation without any gauge conditions enables us to reduce computational costs in case of the conventional one domain problem, and to formally introduce an iterative DDM. However, to the best of our knowledge, mathematical justifications of the numerical scheme without any gauge conditions such as unique solvability and convergency are not available. Therefore, we introduce a gauge condition and formulate magnetic field problems by mixed variational problems. The mixed formulation regards the magnetic vector potential and the Lagrange multiplier as two unknown functions that are approximated by the Nedelec curl-conforming edge element and by the conventional piece wise linear element, respectively. Then, we can introduce a new iterative DDM and can establish ones unique solvability and convergency. Moreover, we can reduce the iterative DDM into more efficient one by using the property of the Lagrange multiplier: in our formulations, the Lagrange multiplier vanishes in the whole domain.Finally, some numerical results are shown in case of ultra-large computational models whose numbers of degrees of freedom are 10E7--10E9. By using the iterative DDM with the gauge condition, we can show efficient convergence results at the iterative procedure to solve the resultant linear systems derived from the artificial interface problem. Moreover, we can confirm that the approximate magnetic vector potential satisfies the gauge condition in the weak sense when we obtain the convergence results.