Numerical Analysis for the Heat Flux in a Mixed Elliptic Problem to Obtain a Discrete Steady-State Two-Phase Stefan Problem
Domingo Alberto Tarzia · SIAM Journal on Numerical Analysis · 1996
We consider a material $\Omega \subset \mathbb{R}^n $ which occupies a convex polygonal bounded domain with regular boundary $\Gamma = \Gamma _1 \cup \Gamma _2 $ (with $\mathop {\Gamma _1 }\limits^ \circ \cap \mathop {\Gamma _2 }\limits^ \circ = \phi $) with meas $(\Gamma _1 ) = | {\Gamma _1 } | > 0$ and $| {\Gamma _2 } | > 0$. We assume, without loss of generality, that the melting temperature is $0^ \circ $C. We apply a temperature $b = {\text{const}} > 0$ on $\Gamma _1 $ and a heat flux $q = {\text{const}} > 0$ 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 $.