A coupled problem arising in induction heating furnaces is studied. The thermal problem, which involves a change of phase, has a nonlocal radiation boundary condition. Convective heat transfer in the liquid is also included which makes necessary to compute the liquid motion. For the space discretization, we propose finite element methods which are combined with characteristics methods in the thermal and flow models to handle the convective terms. In the electromagnetic model they are coupled with boundary element methods (BEM/FEM). An iterative algorithm is introduced for the whole coupled model and numerical results for an industrial induction furnace are presented. © 2010 Elsevier Ltd.