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Convergence of the finite element method applied to an anisotropic phase-field model

Erik Burman[1]; Daniel Kessler[2]; Jacques Rappaz[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

Annales mathématiques Blaise Pascal (2004)

  • Volume: 11, Issue: 1, page 67-94
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Burman, Erik, Kessler, Daniel, and Rappaz, Jacques. "Convergence of the finite element method applied to an anisotropic phase-field model." Annales mathématiques Blaise Pascal 11.1 (2004): 67-94. <http://eudml.org/doc/10500>.

@article{Burman2004,
abstract = {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.},
affiliation = {Ecole Polytechnique Federale Institute of Analysis and Scientific Computing CH-1015 Lausanne Switzerland; University of Maryland Department of Mathematics College Park MD 20740 USA; Ecole Polytechnique Federale Institute of Analysis and Scientific Computing CH-1015 Lausanne Switzerland},
author = {Burman, Erik, Kessler, Daniel, Rappaz, Jacques},
journal = {Annales mathématiques Blaise Pascal},
language = {eng},
month = {1},
number = {1},
pages = {67-94},
publisher = {Annales mathématiques Blaise Pascal},
title = {Convergence of the finite element method applied to an anisotropic phase-field model},
url = {http://eudml.org/doc/10500},
volume = {11},
year = {2004},
}

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
DA - 2004/1//
PB - Annales mathématiques Blaise Pascal
VL - 11
IS - 1
SP - 67
EP - 94
AB - 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.
LA - eng
UR - http://eudml.org/doc/10500
ER -

References

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  1. E. Burman, J. Rappaz, Existence of solutions to an anisotropic phase-field model, Math. Methods Appl. Sci. 26 (2003), 1137-1160 Zbl1032.35053MR1994669
  2. X. Chen, C. M. Elliott, A. Gardiner, J. J. Zhao, Convergence of numerical solutions to the Allen-Cahn equation, Appl. Anal. 69 (1998), 47-56 Zbl0992.65096MR1708186
  3. Z. Chen, K.-H. Hoffmann, An error estimate for a finite-element scheme for a phase field model, IMA J. Numer. Anal. 14 (1994), 243-255 Zbl0801.65091MR1268994
  4. B. Dacorogna, Direct methods in the calculus of variations, 78 (1989), Springer-Verlag, Berlin Zbl0703.49001MR990890
  5. X. Feng, 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 (electronic) Zbl1115.76049MR2028419
  6. D. Kessler, Modeling, mathematical and numerical study of a solutal phase-field model, (2001) 
  7. D. Kessler, O. Krüger, J.F. Scheid, Modeling, mathematical and numerical study of a solutal phase-field model, (1998) 
  8. D. Kessler, 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 Zbl1001.76057MR1897410
  9. R. Kobayashi, A numerical approach to three-dimensional dendritic solidification, Experiment. Math. 3 (1994), 59-81 Zbl0811.65126MR1302819
  10. J. Rappaz, J. F. Scheid, Existence of solutions to a phase-field model for the isothermal solidification process of a binary alloy, Math. Methods Appl. Sci. 23 (2000), 491-513 Zbl0964.35026MR1748319
  11. A. Schmidt, K. G. Siebert, ALBERT—software for scientific computations and applications, Acta Math. Univ. Comenian. (N.S.) 70 (2000), 105-122 Zbl0993.65134MR1865363
  12. J. A. Warren, W. J. Boettinger, Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field model, Acta Metall. 43 (1995), 689-703 

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