A Finite-Element Approximation of Stefan Problems in Heterogeneous Media
Tomáš Roubı́ček · Birkhäuser Basel eBooks · 1990
We will deal with the nonlinear heat transfer equation in media that are only piecewise homogeneous. Let Ω⊂ℝ n , n≤3, be a bounded polyhedral domain with a Lipschitz boundary Γ, covered by a finite number of disjoint polyhedral subdomains Ω i , i=1,...,m; that means Ω i ⊂Ω, Ω i ∩Ω j =ø for i j, $$\underset{i=1}{\overset{m}{\mathop U}}\,{{\overset{-}{\mathop \Omega }\,}_{i}}=\overset{-}{\mathop \Omega }\,$$ , the bar denotes the closure. The nonlinear heat-transfer equation will have in each subdomain Ω i its own (temperature-dependent) coefficients of heat capacity c i =c i (θ) and thermal conductivity k i =k i (θ), θ is the temperature; in other words, the medium is heterogeneous, being composed from m materials occupying respectively the subdomains Ω i , i=1,...m. We also admit c i containing Dirac distributions, by which the Stefan problem in the i th subdomain is modelled. We write here the equations directly in the enthalpy formulation (for the original formulation in terms of temperature we refer to [4,6]). For T>O, α i ,β i :ℝ→ℝ , we consider on each subdomain Ω i the following evolution problem: 1 $$ \left. \begin{array}{l} \theta \left( {x,t} \right) = {\alpha _i}\left( {w\left( {x,t} \right)} \right)\\ \frac{{\partial w}}{{\partial t}} = \Delta {\beta _i}\left( w \right) \end{array} \right\}{\rm{ on }}{Q_i} = {\Omega _i} \times \left( {O,T} \right). $$ where w = w(x,t) is an unknown enthalpy, α i (w) is the temperature and β i (w) is the temperature after the Kirchhoff transformation, we will say briefly the Kirchhoff temperature (β i° α −1 is the kirchhoff transformation, α −1 is the so-called enthalpy transformation).