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Stability and convergence of two discrete schemes for a degenerate solutal non-isothermal phase-field model. (English) Zbl 1171.80006
The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature-dependent properties. It is based on a highly non-linear parabolic system of partial differential equations with three dependent variables: phase-field, solute concentration and temperature. The authors 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 the problem. The first scheme is nonlinear, unconditionally stable and convergent. Whereas the second scheme is linear but conditionally stable and convergent.

##### MSC:
 80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 80A22 Stefan problems, phase changes, etc. 35K65 Degenerate parabolic equations 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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##### References:
 [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 · Zbl 1012.35049 [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 · Zbl 1117.80002 [3] S. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathathematics15. Springer-Verlag, Berlin (1994). Zbl0804.65101 · Zbl 0804.65101 [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 · Zbl 1155.74404 [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). · Zbl 1059.65103 [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 · Zbl 1115.76049 [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 · Zbl 1193.35153 [9] O. Kavian, Introduction à la Théorie des Points Critiques, Mathématiques et Applications13. Springer, Berlin (1993). Zbl0797.58005 · Zbl 0797.58005 [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 · Zbl 1001.76057 [11] R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp.38 (1982) 437-445. Zbl0483.65007 · Zbl 0483.65007 [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 · Zbl 1073.35136 [13] J. Simon, Compact sets in the Space Lp(0,T;B). Ann. Mat. Pura Appl.146 (1987) 65-97. Zbl0629.46031 · Zbl 0629.46031
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