Well posed reconstruction of the solar coronal magnetic field

We present and compare two methods for the reconstruction of the solar coronal magnetic field, assumed to be force-free, from photospheric boundary data. Both methods rely on a well posed mathematical boundary value problem and are of the Grad-Rubin type, i.e., the couple $({\vec B},\alpha)$ is computed iteratively. They do differ from each other on the one hand by the way they address the zero-divergence of ${\vec B}$ issue, and on the other hand by the scheme they use for computing  α at each iteration. The comparison of the two methods is done by numerically computing two examples of nonlinear force-free fields associated to large scale strong electric current distributions, whose exact forms can be otherwise determined semi-analytically. In particular, the second solution has a large nonlinearity even in the weak field region – a feature which is not present in the actual magnetograms, but is interesting to consider as it does allow to push the methods to the limits of their range of validity. The best results obtained with those methods give a relative vector error smaller than 0.01. For the latter extreme case, our results show that higher resolution reconstructions with bounded convergence improve the approximated solution, which may be of some interest for the treatment of the data of future magnetographs.