A nonlinear boundary value problem solved by spectral methods

We study a nonlinear boundary value problem posed in the interior or the exterior of the unit disk in R2. Using capacity operator, we transform it into a pseudo-differential problem on the unit circle. The Galerkin method together with Fourier expansion, is used to approximate our problem. We show the convergence of the fixed-point scheme and we give an accurate bound of the L2-norm of the error. Numerical results coming from a problem arising in electromagnetic casting are also presented.