Stability of flat interfaces during semidiscrete solidification
ESAIM: Mathematical Modelling and Numerical Analysis (2010)
- Volume: 36, Issue: 4, page 573-595
- ISSN: 0764-583X
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topVeeser, Andreas. "Stability of flat interfaces during semidiscrete solidification." ESAIM: Mathematical Modelling and Numerical Analysis 36.4 (2010): 573-595. <http://eudml.org/doc/194117>.
@article{Veeser2010,
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.
},
author = {Veeser, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {(Mullins-Sekerka) stability analysis; morphological instabilities;
spatial semidiscretization; moving finite elements; phase
transitions; surface tension; Stefan condition; dendritic growth;
secondary sidebranching.; Mullins-Sekerka stability analysis; spatial semidiscretization; phase transitions; secondary sidebranching},
language = {eng},
month = {3},
number = {4},
pages = {573-595},
publisher = {EDP Sciences},
title = {Stability of flat interfaces during semidiscrete solidification},
url = {http://eudml.org/doc/194117},
volume = {36},
year = {2010},
}
TY - JOUR
AU - Veeser, Andreas
TI - Stability of flat interfaces during semidiscrete solidification
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 4
SP - 573
EP - 595
AB -
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.
LA - eng
KW - (Mullins-Sekerka) stability analysis; morphological instabilities;
spatial semidiscretization; moving finite elements; phase
transitions; surface tension; Stefan condition; dendritic growth;
secondary sidebranching.; Mullins-Sekerka stability analysis; spatial semidiscretization; phase transitions; secondary sidebranching
UR - http://eudml.org/doc/194117
ER -
References
top- 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.
- X. Chen, J. Hong and F. Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations21 (1996) 1705-1727.
- K. Deckelnick and G. Dziuk, Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math.72 (1995) 197-222.
- J. Escher and G. Simonett, Classical solutions for Hele-Shaw models with surface tension. Adv. Differential Equations2 (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).
- J.S. Langer, Instabilities and pattern formation in crystal growth. Rev. Modern Phys.52 (1980) 1-28.
- 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.
- L. Perko, Differential equations and dynamical systems. 2nd ed, Vol. 7 of Texts in Applied Mathematics. Springer, New York (1996).
- A. Schmidt, Computation of three dimensional dendrites with finite elements. J. Comput. Phys.125 (1996) 293-312.
- 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.
- J. Strain, Velocity effects in unstable solidification. SIAM J. Appl. Math.50 (1990) 1-15.
- 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).
- A. Veeser, Error estimates for semi-discrete dendritic growth. Interfaces Free Bound.1 (1999) 227-255.
- 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).
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