Numerical analysis of the multidimensional Stefan problem with supercooling and superheating. (English) Zbl 0629.65130

Let \(\Omega\) be a bounded domain in \(R^ n\) (N\(\geq 1)\) occupied by water and ice, \(Q=\Omega \times (0,T)\); denote the relative temperature by \(\theta\), the water concentration by \(\chi\) \((\chi =1\) in the water phase, \(\chi =0\) in the ice phase, \(0<\chi <1\) in the mushy region). In the standard Stefan model the constitutive equation \((1)\quad C_ p \partial \theta /\partial t+L \partial \chi /\partial t-k\Delta \theta =f\) in D’(Q) is complemented by the equilibrium condition \((2)\quad \chi \in H(\theta)\) in Q, where H denotes the Heaviside graph \((H(x)=\{0\}\) if \(x<0\), \(H(x)=\{1\}\) if \(x>0\), \(H(0)=[0,1])\). In this paper, supercooling and superheating effects are taken into account by replacing (2) by a relaxation law of the form \((3)\quad \alpha \partial \chi /\partial t+H^{-1}(\chi)\ni \beta (\theta,\chi)\) in Q with \(\alpha >0\) and \(\beta \in C^ 0(R\times [0,1])\), \(\beta (0,\chi)=0\), \(\beta\) (\(\cdot,\chi)\) strictly increasing in a neighbourhood of 0 for all \(\chi\in [0,1].\)
In section 1 of the paper a variational formulation of the problem (1), (3) is introduced and results on existence, uniqueness and regularity of the solution are given. In section 2 the continuous problem is discretized by means of a semi-implicit scheme in time and finite elements (linear for \(\theta\), constant for \(\chi)\) in space; in section 3 error estimates are proved for \(\theta\) in \(L^ 2(0,T;L^ 2(\Omega))\) and for \(\chi\) in \(L^{\infty}(0,T;L^ 2(\Omega))\); they turn out to be stronger than those for the usual Stefan problem.
Reviewer: W.Müller


65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
80A17 Thermodynamics of continua