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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 (B, α) is computed iteratively. They do differ from each other on the one hand by the way they address the zero-divergence of 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.

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Non-linear force-free fields (or Beltrami fields) are three-component divergence-free fields solutions of the equation curl B × B = 0. The aim of this paper is to prove existence of solutions of a corresponding boundary value problem in a simply or multiply connected domain of R 3 . The proof is based on a fixed point procedure coupled with a singular perturbation skill. Mathematics Subject Classification (2000). 35F30, 35M10, 35Q35, 35Q72, 85A30, 76Cxx.

SUMMARYThe purpose of the present paper is twofold. The ÿrst object is to study the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half-space of R N . The behaviour of solutions at inÿnity is described by means of a family of weighted Sobolev spaces. A class of existence, uniqueness and regularity results are obtained. The second purpose is to investigate some properties of grad, div and curl operators in order to treat curl-div systems of the form curl w = u; div w = 0 and problems related to vector potentials and Helmholtz decomposition.

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Abstract.In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of R n . The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.Mathematics Subject Classification. 35J, 35J05, 65Jxx, 65Nxx, 65Rxx.

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