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Stability and convergence of two discrete schemesfor a degenerate solutal non-isothermalphase-field model

Published online by Cambridge University Press:  30 April 2009

Francisco Guillén-González  and
Juan Vicente Gutiérrez-Santacreu
Francisco Guillén-González
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. guillen@us.es; juanvi@us.es
Juan Vicente Gutiérrez-Santacreu
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. guillen@us.es; juanvi@us.es
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Abstract

We analyze two numerical schemes of Euler type in time and C 0finite-element type with $\mathbb{P}_1$ -approximation in space forsolving a phase-field model of a binary alloy with thermalproperties. This model is written as a highly non-linear parabolicsystem with three unknowns: phase-field, solute concentration andtemperature, where the diffusion for the temperature and soluteconcentration may degenerate. The first scheme is nonlinear, unconditionally stableand convergent. The other scheme is linear but conditionally stableand 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. Inaddition, for the model when the heat conductivity and solutediffusion coefficients are constants, optimal error estimates forboth schemes are shown based on stability estimates.

Keywords

Phase-field modelsdiffuse interface modelsolidification processdegenerate parabolic systemsbackward Euler schemesfinite elementsstabilityconvergenceerror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 563 - 589
Copyright
© EDP Sciences, SMAI, 2009

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