Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model

Guillén-González, Francisco; Gutiérrez-Santacreu, Juan Vicente

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 563-589.

We 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 a binary alloy with thermal properties. This model is written as a highly non-linear parabolic system with three unknowns: phase-field, solute concentration and temperature, where the diffusion for the temperature and solute concentration may degenerate. The first scheme is nonlinear, unconditionally stable and convergent. The other scheme is linear but conditionally stable and convergent. A maximum principle is avoided in both schemes, using a truncation operator on the $L^2$ projection onto the ${\mathbb{P}}_0$ finite element for the discrete concentration. In addition, for the model when the heat conductivity and solute diffusion coefficients are constants, optimal error estimates for both schemes are shown based on stability estimates.

Classification : 35Q72, 35K65, 65M12, 65M60
Mots clés : phase-field models, diffuse interface model, solidification process, degenerate parabolic systems, backward Euler schemes, finite elements, stability, convergence, error estimates

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