Convergence of the finite element method applied to an anisotropic phase-field model
Annales mathématiques Blaise Pascal, Volume 11 (2004) no. 1, p. 67-94

We formulate a finite element method for the computation of solutions to an anisotropic phase-field model for a binary alloy. Convergence is proved in the H 1 -norm. The convergence result holds for anisotropy below a certain threshold value. We present some numerical experiments verifying the theoretical results. For anisotropy below the threshold value we observe optimal order convergence, whereas in the case where the anisotropy is strong the numerical solution to the phase-field equation does not converge.

@article{AMBP_2004__11_1_67_0,
     author = {Burman, Erik and Kessler, Daniel and Rappaz, Jacques},
     title = {Convergence of the finite element method applied to an anisotropic phase-field model},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {11},
     number = {1},
     year = {2004},
     pages = {67-94},
     doi = {10.5802/ambp.186},
     mrnumber = {2077239},
     zbl = {02207859},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2004__11_1_67_0}
}
Burman, Erik; Kessler, Daniel; Rappaz, Jacques. Convergence of the finite element method applied to an anisotropic phase-field model. Annales mathématiques Blaise Pascal, Volume 11 (2004) no. 1, pp. 67-94. doi : 10.5802/ambp.186. http://www.numdam.org/item/AMBP_2004__11_1_67_0/

[1] Burman, E.; Rappaz, J. Existence of solutions to an anisotropic phase-field model, Math. Methods Appl. Sci., Tome 26 (2003) no. 13, pp. 1137-1160 | Article | MR 1994669 | Zbl 1032.35053

[2] Chen, X.; Elliott, C. M.; Gardiner, A.; Zhao, J. J. Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal., Tome 69 (1998) no. 1-2, pp. 47-56 | MR 1708186 | Zbl 0992.65096

[3] Chen, Z.; Hoffmann, K.-H. An error estimate for a finite-element scheme for a phase field model, IMA J. Numer. Anal., Tome 14 (1994) no. 2, pp. 243-255 | Article | MR 1268994 | Zbl 0801.65091

[4] Dacorogna, B. Direct methods in the calculus of variations, Springer-Verlag, Berlin, Applied Mathematical Sciences, Tome 78 (1989) | MR 990890 | Zbl 0703.49001

[5] Feng, X.; Prohl, A. Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits, Math. Comp., Tome 73 (2004) no. 246, p. 541-567 (electronic) | Article | MR 2028419 | Zbl 1115.76049 | Zbl 02041046

[6] Kessler, D. Modeling, mathematical and numerical study of a solutal phase-field model, Mathematics department, EPFL (2001) (Ph. D. Thesis)

[7] Kessler, D.; Krüger, O.; Scheid, J.F. Modeling, mathematical and numerical study of a solutal phase-field model (1998) (PREPRINT no. 10.98, DMA, EPFL)

[8] Kessler, D.; 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., Tome 22 (2002) no. 2, pp. 281-305 | Article | MR 1897410 | Zbl 1001.76057

[9] Kobayashi, R. A numerical approach to three-dimensional dendritic solidification, Experiment. Math., Tome 3 (1994) no. 1, pp. 59-81 | MR 1302819 | Zbl 0811.65126

[10] Rappaz, J.; Scheid, J. F. Existence of solutions to a phase-field model for the isothermal solidification process of a binary alloy, Math. Methods Appl. Sci., Tome 23 (2000) no. 6, pp. 491-513 | Article | MR 1748319 | Zbl 0964.35026

[11] Schmidt, A.; Siebert, K. G. ALBERT—software for scientific computations and applications, Acta Math. Univ. Comenian. (N.S.), Tome 70 (2000) no. 1, pp. 105-122 | MR 1865363 | Zbl 0993.65134

[12] Warren, J. A.; Boettinger, W. J. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field model, Acta Metall., Tome 43 (1995), pp. 689-703 | Article