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Error estimates for two-phase Stefan problems in several space variables. I: Linear boundary conditions. (English) Zbl 0606.65084
The enthalpy formulation of two-phase Stefan problems, with linear boundary conditions, is approximated by \(C^ 0\)-piecewise linear finite elements in space and backward-differences in time combined with a regularization procedure. Error estimates of \(L^ 2\)-type are obtained. For general regularized problems an order \(\epsilon^{1/2}\) is proved, while the order is shown to be \(\epsilon\) for non-degenerate cases. For discrete problems an order \(h^ 2\epsilon^{-1}+h+\tau \epsilon^{- }+\tau^{2/3}\) is obtained. These orders impose the relations \(\epsilon \sim \tau \sim h^{4/3}\) for the general case and \(\epsilon \sim h\sim \tau^{2/3}\) for non-degenerate problems, in order to obtain rates of convergence \(h^{2/3}\) or h respectively. Besides, an order \(h| \log h| +\tau^{1/2}\) is shown for finite element meshes with certain approximation property. Also continuous dependence of discrete solutions upon the data is proved.

MSC:
65Z05 Applications to the sciences
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K05 Heat equation
80A17 Thermodynamics of continua
35R35 Free boundary problems for PDEs
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