## 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

### MSC:

 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