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This paper concerns the mathematical modelling and numerical solution of thermoelectrical phenomena taking place in an axisymmetric induction heating furnace. We formulate the problem in a two-dimensional domain and propose a finite element method and an iterative algorithm for its numerical solution. We also provide a family of one-dimensional analytical solutions which are used to test the two-dimensional code and to predict the behaviour of the furnace under special conditions. Some numerical results for an industrial furnace used in silicon purification are shown.

Secondly, it is necessary to give a precise mathematical definition of the termappearing in Equation (19), because it does not make sense for any G ∈ Y. To attain this goal, for k = 1, . . . , m, let us denote by k the solution in H 1 ( k \ k ), unique up to a constant, of the following problem:). Since the current density J = E satisfies div J = 0 in and J · n = 0 on , we havewhere k denotes the boundary of k . Note that, while the right-hand side of the previous equality does not make sense for any G in Y, the left-hand side does.

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