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Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model. (English) Zbl 1171.80006
The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature-dependent properties. It is based on a highly non-linear parabolic system of partial differential equations with three dependent variables: phase-field, solute concentration and temperature. The authors analyze two numerical schemes of Euler type in time and \(C^0\) finite-element type with \(\mathbb{P}_1\)-approximation in space for solving a phase-field model of the problem. The first scheme is nonlinear, unconditionally stable and convergent. Whereas the second scheme is linear but conditionally stable and convergent.

MSC:
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
80A22 Stefan problems, phase changes, etc.
35K65 Degenerate parabolic equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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