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.

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.

DOI: 10.5802/ambp.186
Burman, Erik 1; Kessler, Daniel 2; Rappaz, Jacques 1

1 Ecole Polytechnique Federale Institute of Analysis and Scientific Computing CH-1015 Lausanne Switzerland
2 University of Maryland Department of Mathematics College Park MD 20740 USA
@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},
pages = {67--94},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {11},
number = {1},
year = {2004},
doi = {10.5802/ambp.186},
mrnumber = {2077239},
zbl = {02207859},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ambp.186/}
}
TY  - JOUR
AU  - Burman, Erik
AU  - Kessler, Daniel
AU  - Rappaz, Jacques
TI  - Convergence of the finite element method applied to an anisotropic phase-field model
JO  - Annales mathématiques Blaise Pascal
PY  - 2004
DA  - 2004///
SP  - 67
EP  - 94
VL  - 11
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - http://archive.numdam.org/articles/10.5802/ambp.186/
UR  - https://www.ams.org/mathscinet-getitem?mr=2077239
UR  - https://zbmath.org/?q=an%3A02207859
UR  - https://doi.org/10.5802/ambp.186
DO  - 10.5802/ambp.186
LA  - en
ID  - AMBP_2004__11_1_67_0
ER  - 
%0 Journal Article
%A Burman, Erik
%A Kessler, Daniel
%A Rappaz, Jacques
%T Convergence of the finite element method applied to an anisotropic phase-field model
%J Annales mathématiques Blaise Pascal
%D 2004
%P 67-94
%V 11
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.186
%R 10.5802/ambp.186
%G en
%F 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://archive.numdam.org/articles/10.5802/ambp.186/

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

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

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

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

[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., Volume 73 (2004) no. 246, p. 541-567 (electronic) | DOI | MR | Zbl

[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., Volume 22 (2002) no. 2, pp. 281-305 | DOI | MR | Zbl

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

[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., Volume 23 (2000) no. 6, pp. 491-513 | DOI | MR | Zbl

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

[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., Volume 43 (1995), pp. 689-703 | DOI

Cited by Sources: