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.

Classification: 65M12, 65M60

Keywords: (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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {36}, number = {4}, year = {2002}, pages = {573-595}, doi = {10.1051/m2an:2002026}, zbl = {1137.65404}, mrnumber = {1932305}, language = {en}, url = {http://www.numdam.org/item/M2AN_2002__36_4_573_0} }

Stability of flat interfaces during semidiscrete solidification. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 4, pp. 573-595. doi : 10.1051/m2an:2002026. http://www.numdam.org/item/M2AN_2002__36_4_573_0/

[1] Mathematical modeling of melting and freezing processes. Hemisphere Publishing Corporation, Washington (1993).

and ,[2] Ordinary differential equations. An introduction to nonlinear analysis, Vol. 13 of De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (1990). | MR 1071170 | Zbl 0708.34002

,[3] A finite element method for dendritic growth, in Computational crystal growers workshop, J.E. Taylor Ed., AMS Selected Lectures in Mathematics (1992) 16-20.

and ,[4] Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations 21 (1996) 1705-1727. | Zbl 0884.35177

, and ,[5] Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math. 72 (1995) 197-222. | Zbl 0838.65103

and ,[6] Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations 2 (1997) 619-642. | Zbl 1023.35527

and ,[7] 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 0880.35140

and ,[8] Measure Theory and Fine Properties of Functions. CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Ratin, Stud. Adv. Math., 33431, Florida (1992). | MR 1158660 | Zbl 0804.28001

and ,[9] A level set based finite element algorithm for the simulation of dendritic growth. Submitted to Computing and Visualization in Science, Springer. | Zbl 1120.80310

,[10] Thermomechanics of evolving phase boundaries in the plane. Clarendon Press, Oxford (1993). | MR 1402243 | Zbl 0787.73004

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

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

and ,[13] Differential equations and dynamical systems. 2nd ed, Vol. 7 of Texts in Applied Mathematics. Springer, New York (1996). | MR 1418638 | Zbl 0854.34001

,[14] Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125 (1996) 293-312. | Zbl 0844.65096

,[15] 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 0563.73100

,[16] Velocity effects in unstable solidification. SIAM J. Appl. Math. 50 (1990) 1-15. | Zbl 0698.35166

,[17] An analysis of the finite element method. Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J (1973). | MR 443377 | Zbl 0356.65096

and ,[18] Error estimates for semi-discrete dendritic growth. Interfaces Free Bound. 1 (1999) 227-255. | Zbl 0952.35158

,[19] Models of phase transitions, Vol. 28 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1996). | MR 1423808 | Zbl 0882.35004

,[20] Weakly Differentiable Functions, Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York (1989). | MR 1014685 | Zbl 0692.46022

,