Numerical analysis for the heat flux in a mixed elliptic problem to obtain a discrete steady-state two-phase Stefan problem

We consider a material Omega subset of R(n) which occupies a convex polygonal bounded domain with regular boundary Gamma = Gamma(1) boolean OR Gamma(2) (with Gamma(1) boolean AND Gamma(2) = 0) with meas(Gamma(1)) = \Gamma(1)\ > 0 and \Gamma(2)\ > 0. We assume, without loss of generality, that the melting temperature is 0 degrees C. We apply a temperature b = Const > 0 on Gamma(1) and a heat flux q = Const > 0 on Gamma(2). We consider a steady-state heat conduction problem in Omega. We consider a regular triangulation of the domain Omega with Lagrange triangles of type 1. We study sufficient (and/or necessary) conditions on the heat flux q on Gamma(2) to obtain a change of phase (steady-state, two-phase, discretized Stefan problem) in the corresponding discretized domain, that is, a discrete temperature of nonconstant sign in Omega.