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Stability of flat interfaces during semidiscrete solidification

Published online by Cambridge University Press:  15 September 2002

Andreas Veeser
Andreas Veeser*
Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy. veeser@mat.unimi.it. (On leave from) Institut für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg, Germany. andy@mathematik.uni-freiburg.de.
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Abstract

The stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the Stefan problem with dynamical Gibbs–Thomson law. The stability analysis bases on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed.

Keywords

(Mullins-Sekerka) stability analysismorphological instabilitiesspatial semidiscretizationmoving finite elementsphase transitionssurface tensionStefan conditiondendritic growthsecondary sidebranching.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 573 - 595
Copyright
© EDP Sciences, SMAI, 2002

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References

V. Alexiades and A.D. Solomon, Mathematical modeling of melting and freezing processes. Hemisphere Publishing Corporation, Washington (1993).
H. Amann, Ordinary differential equations. An introduction to nonlinear analysis, Vol. 13 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (1990).
E. Bänsch and A. Schmidt, A finite element method for dendritic growth, in Computational crystal growers workshop, J.E. Taylor Ed., AMS Selected Lectures in Mathematics (1992) 16-20.
Chen, X., Hong, J. and Existence, F. Yi, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations 21 (1996) 1705-1727.
Deckelnick, K. and Dziuk, G., Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math. 72 (1995) 197-222. CrossRef
Escher, J. and Simonett, G., Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2 (1997) 619-642.
J. Escher and G. Simonett, Classical solutions for the quasi-stationary Stefan problem with surface tension, in Papers associated with the international conference on partial differential equations, Potsdam, Germany, June 29-July 2, 1996, M. Demuth et al. Eds., Vol. 100. Akademie Verlag, Math. Res., Berlin (1997) 98-104.
L.C. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Ratin, Stud. Adv. Math., 33431, Florida (1992).
M. Fried, A level set based finite element algorithm for the simulation of dendritic growth. Submitted to Computing and Visualization in Science, Springer.
M.E. Gurtin, Thermomechanics of evolving phase boundaries in the plane. Clarendon Press, Oxford (1993).
Langer, J.S., Instabilities and pattern formation in crystal growth. Rev. Modern Phys. 52 (1980) 1-28. CrossRef
Mullins, W.W. and Sekerka, R.F., Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35 (1964) 444-451. CrossRef
L. Perko, Differential equations and dynamical systems. 2nd ed, Vol. 7 of Texts in Applied Mathematics. Springer, New York (1996).
Schmidt, A., Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125 (1996) 293-312. CrossRef
R.F. Sekerka, Morphological instabilities during phase transformations, in Phase transformations and material instabilities in solids, Proc. Conf., Madison/Wis. 1983. Madison 52, M. Gurtin Ed., Publ. Math. Res. Cent. Univ. Wis. (1984) 147-162.
Strain, J., Velocity effects in unstable solidification. SIAM J. Appl. Math. 50 (1990) 1-15. CrossRef
G. Strang and G.J. Fix, An analysis of the finite element method. Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J (1973).
Veeser, A., Error estimates for semi-discrete dendritic growth. Interfaces Free Bound. 1 (1999) 227-255. CrossRef
A. Visintin, Models of phase transitions, Vol. 28 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1996).
W.P. Ziemer, Weakly Differentiable Functions, Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989).