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

Francisco Guillén-González; Juan Vicente Gutiérrez-Santacreu

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

- Volume: 43, Issue: 3, page 563-589
- ISSN: 0764-583X

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topGuillén-González, Francisco, and Gutiérrez-Santacreu, Juan Vicente. "Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model." ESAIM: Mathematical Modelling and Numerical Analysis 43.3 (2009): 563-589. <http://eudml.org/doc/250594>.

@article{Guillén2009,

abstract = {
We analyze two numerical schemes of Euler type in time and C0
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 L2 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.
},

author = {Guillén-González, Francisco, Gutiérrez-Santacreu, Juan Vicente},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Phase-field models; diffuse interface model; solidification process; degenerate parabolic systems; backward Euler schemes; finite elements; stability; convergence; error estimates.; phase-field models; error estimates},

language = {eng},

month = {4},

number = {3},

pages = {563-589},

publisher = {EDP Sciences},

title = {Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model},

url = {http://eudml.org/doc/250594},

volume = {43},

year = {2009},

}

TY - JOUR

AU - Guillén-González, Francisco

AU - Gutiérrez-Santacreu, Juan Vicente

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2009/4//

PB - EDP Sciences

VL - 43

IS - 3

SP - 563

EP - 589

AB -
We analyze two numerical schemes of Euler type in time and C0
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 L2 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.

LA - eng

KW - Phase-field models; diffuse interface model; solidification process; degenerate parabolic systems; backward Euler schemes; finite elements; stability; convergence; error estimates.; phase-field models; error estimates

UR - http://eudml.org/doc/250594

ER -

## References

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- D. Kessler and J.F. Scheid, 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) 281–305. Zbl1001.76057
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