no. 4
Stability of flat interfaces during semidiscrete solidification
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 573-595.

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.

DOI : 10.1051/m2an:2002026
Classification : 65M12, 65M60
Mots clés : (Mullins-Sekerka) stability analysis, morphological instabilities, spatial semidiscretization, moving finite elements, phase transitions, surface tension, Stefan condition, dendritic growth, secondary sidebranching 
@article{M2AN_2002__36_4_573_0,
     author = {Veeser, Andreas},
     title = {Stability of flat interfaces during semidiscrete solidification},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {573--595},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {4},
     year = {2002},
     doi = {10.1051/m2an:2002026},
     mrnumber = {1932305},
     zbl = {1137.65404},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2002026/}
}
TY  - JOUR
AU  - Veeser, Andreas
TI  - Stability of flat interfaces during semidiscrete solidification
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2002
SP  - 573
EP  - 595
VL  - 36
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2002026/
DO  - 10.1051/m2an:2002026
LA  - en
ID  - M2AN_2002__36_4_573_0
ER  - 
%0 Journal Article
%A Veeser, Andreas
%T Stability of flat interfaces during semidiscrete solidification
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2002
%P 573-595
%V 36
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2002026/
%R 10.1051/m2an:2002026
%G en
%F M2AN_2002__36_4_573_0
Veeser, Andreas. Stability of flat interfaces during semidiscrete solidification. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 573-595. doi : 10.1051/m2an:2002026. http://www.numdam.org/articles/10.1051/m2an:2002026/

[1] V. Alexiades and A.D. Solomon, Mathematical modeling of melting and freezing processes. Hemisphere Publishing Corporation, Washington (1993).

[2] H. Amann, Ordinary differential equations. An introduction to nonlinear analysis, Vol. 13 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (1990). | MR | Zbl

[3] 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.

[4] X. Chen, J. Hong and F. Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations 21 (1996) 1705-1727. | Zbl

[5] K. Deckelnick and G. Dziuk, Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math. 72 (1995) 197-222. | Zbl

[6] J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2 (1997) 619-642. | Zbl

[7] 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. | Zbl

[8] 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). | MR | Zbl

[9] M. Fried, A level set based finite element algorithm for the simulation of dendritic growth. Submitted to Computing and Visualization in Science, Springer. | Zbl

[10] M.E. Gurtin, Thermomechanics of evolving phase boundaries in the plane. Clarendon Press, Oxford (1993). | MR | Zbl

[11] J.S. Langer, Instabilities and pattern formation in crystal growth. Rev. Modern Phys. 52 (1980) 1-28.

[12] W.W. Mullins and R.F. Sekerka, Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 35 (1964) 444-451.

[13] L. Perko, Differential equations and dynamical systems. 2nd ed, Vol. 7 of Texts in Applied Mathematics. Springer, New York (1996). | MR | Zbl

[14] A. Schmidt, Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125 (1996) 293-312. | Zbl

[15] 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. | Zbl

[16] J. Strain, Velocity effects in unstable solidification. SIAM J. Appl. Math. 50 (1990) 1-15. | Zbl

[17] 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). | MR | Zbl

[18] A. Veeser, Error estimates for semi-discrete dendritic growth. Interfaces Free Bound. 1 (1999) 227-255. | Zbl

[19] A. Visintin, Models of phase transitions, Vol. 28 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1996). | MR | Zbl

[20] W.P. Ziemer, Weakly Differentiable Functions, Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989). | MR | Zbl

Cité par Sources :