An iterative method for the reconstruction of the solar coronal magnetic field: I. Method for regular solutions

We present a method for reconstructing the coronal magnetic field, assumed to be in a non-linear force-free state, from its values given in the photosphere by vector magnetograph measurements. In this paper, that is the first of a series, we propose a method that solves the boundary value problem set in the functional space of regular solutions (i.e., that do not contain current sheets). This is an iterative method introduced by Grad and Rubin. It is associated with a well-posed boundary-value problem. We present some results obtained with this method on two exact solutions of the magnetostatic equations, used as theoretical magnetograms. Unlike some other extrapolations methods, that are associated with ill-posed boundary value problems, our method allows extrapolation to arbitrarily large heights, with no blowing up due to the presence in these methods of an intrinsic instability that makes errors growing up exponentially.