Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T20:10:51.639Z Has data issue: false hasContentIssue false

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
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected]; [email protected]
Juan Vicente Gutiérrez-Santacreu
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. [email protected]; [email protected]
Get access

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Boldrini, J.L. and Planas, G., Weak solutions of a phase-field model for phase change of an alloy with thermal properties. Math. Methods Appl. Sci. 25 (2002) 11771193. CrossRef
J.L. Boldrini and C. Vaz, A semidiscretization scheme for a phase-field type model for solidification. Port. Math. (N.S.) 63 (2006) 261–292.
S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathathematics 15. Springer-Verlag, Berlin (1994).
Burman, E., Kessler, D. and Rappaz, J., Convergence of the finite element method applied to an anisotropic phase-field model. Ann. Math. Blaise Pascal 11 (2004) 6794. CrossRef
Caginalp, G. and Xie, W., Phase-field and sharp-interface alloy models. Phys. Rev. E 48 (1993) 18971909. CrossRef
A. Ern and J.L.Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer, New York (2004).
Feng, X. and Prohl, A., Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp. 73 (2004) 541567. CrossRef
Guillén-González, F. and Gutiérrez-Santacreu, J.V., Unconditional stability and convergence of a fully discrete scheme for 2D viscous fluids models with mass diffusion. Math. Comp. 77 (2008) 14951524 (electronic). CrossRef
O. Kavian, Introduction à la Théorie des Points Critiques, Mathématiques et Applications 13. Springer, Berlin (1993).
Kessler, D. and Scheid, J.F., A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy. IMA J. Numer. Anal. 22 (2002) 281305. CrossRef
Rannacher, R. and Scott, R., Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437445. CrossRef
Scheid, J.F., Global solutions to a degenerate solutal phase field model for the solidification of a binary alloy. Nonlinear Anal. 5 (2004) 207217. CrossRef
J. Simon, Compact sets in the Space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–97.