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

Abstract

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We analyze two numerical schemes of Euler type in time and C0 finite-element type with 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 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.

How to cite

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Guillén-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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  1. J.L. Boldrini and G. Planas, Weak solutions of a phase-field model for phase change of an alloy with thermal properties. Math. Methods Appl. Sci.25 (2002) 1177–1193.  Zbl1012.35049
  2. 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.  Zbl1117.80002
  3. S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathathematics15. Springer-Verlag, Berlin (1994).  Zbl0804.65101
  4. E. Burman, D. Kessler and J. Rappaz, Convergence of the finite element method applied to an anisotropic phase-field model. Ann. Math. Blaise Pascal11 (2004) 67–94.  Zbl1155.74404
  5. G. Caginalp and W. Xie, Phase-field and sharp-interface alloy models. Phys. Rev. E48 (1993) 1897–1909.  
  6. A. Ern and J.L.Guermond, Theory and practice of finite elements, Applied Mathematical Sciences159. Springer, New York (2004).  Zbl1059.65103
  7. X. Feng and A. Prohl, Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp.73 (2004) 541–567.  Zbl1115.76049
  8. F. Guillén-González and J.V. Gutiérrez-Santacreu, Unconditional stability and convergence of a fully discrete scheme for 2D viscous fluids models with mass diffusion. Math. Comp.77 (2008) 1495–1524 (electronic).  Zbl1193.35153
  9. O. Kavian, Introduction à la Théorie des Points Critiques, Mathématiques et Applications13. Springer, Berlin (1993).  Zbl0797.58005
  10. 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
  11. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp.38 (1982) 437–445.  Zbl0483.65007
  12. J.F. Scheid, Global solutions to a degenerate solutal phase field model for the solidification of a binary alloy. Nonlinear Anal.5 (2004) 207–217.  Zbl1073.35136
  13. J. Simon, Compact sets in the Space Lp(0,T;B). Ann. Mat. Pura Appl.146 (1987) 65–97.  

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