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Stability of flat interfaces during semidiscrete solidification. (English) Zbl 1137.65404

Summary: 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.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[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 1071170 | Zbl 0708.34002 · Zbl 0708.34002
[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 0884.35177 · Zbl 0884.35177 · doi:10.1080/03605309608821243
[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 0838.65103 · Zbl 0838.65103 · doi:10.1007/s002110050166
[6] J. Escher and G. Simonett , Classical solutions for Hele-Shaw models with surface tension . Adv. Differential Equations 2 ( 1997 ) 619 - 642 . Zbl 1023.35527 · Zbl 1023.35527
[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 0880.35140 · Zbl 0880.35140
[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 1158660 | Zbl 0804.28001 · Zbl 0804.28001
[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 1120.80310 · Zbl 1120.80310 · doi:10.1007/s00791-004-0141-4
[10] M.E. Gurtin , Thermomechanics of evolving phase boundaries in the plane . Clarendon Press, Oxford ( 1993 ). MR 1402243 | Zbl 0787.73004 · Zbl 0787.73004
[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 1418638 | Zbl 0854.34001 · Zbl 0854.34001
[14] A. Schmidt , Computation of three dimensional dendrites with finite elements . J. Comput. Phys. 125 ( 1996 ) 293 - 312 . Zbl 0844.65096 · Zbl 0844.65096 · doi:10.1006/jcph.1996.0095
[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 0563.73100 · Zbl 0563.73100
[16] J. Strain , Velocity effects in unstable solidification . SIAM J. Appl. Math. 50 ( 1990 ) 1 - 15 . Zbl 0698.35166 · Zbl 0698.35166 · doi:10.1137/0150001
[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 443377 | Zbl 0356.65096 · Zbl 0356.65096
[18] A. Veeser , Error estimates for semi-discrete dendritic growth . Interfaces Free Bound. 1 ( 1999 ) 227 - 255 . Zbl 0952.35158 · Zbl 0952.35158 · doi:10.4171/IFB/10
[19] A. Visintin , Models of phase transitions , Vol. 28 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston ( 1996 ). MR 1423808 | Zbl 0882.35004 · Zbl 0882.35004
[20] W.P. Ziemer , Weakly Differentiable Functions , Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag, New York ( 1989 ). MR 1014685 | Zbl 0692.46022 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
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