Numerical approximation of free boundary problem arising in electromagnetic shaping

We approximate numerically the section of a column of liquid metal submitted to an external electromagnetic field. The bidimensional model used here leads us to solve by a spectral method a nonlinear boundary problem posed on the exterior of the unit disk to find a convenient conformal mapping. We give accurates bound of the errors obtained during our approximation process. We present some examples of shapes for both exterior and interior shaping