Abstract
The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.
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References
Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36, 1–37 (2004)
Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)
Attouch, H.: Variational Convergence of Functions and Operators. Pitman, Boston (1984)
Blesgen, T.: A generalisation of the Navier-Stokes equation to two-phase flows. J. Phys. D 32, 1119–1123 (1999)
Braides, A., dal Maso, G., Garroni, A.: Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Rational Mech. Anal. 146, 23–58 (1999)
Braides, A., Gelli, M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9, 363–399 (2002)
Cahn, J.W., Hilliard, J.E.: Free energy of a non-uniform system I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
De Giorgi, E.: Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rend. Mat. 4, 95–113 (1955)
De Giorgi, E., Franzoni, T.: Su un tipo di convergenza variazionali. Atti Accad. Naz. Lincei 8, 277–294 (1975)
Friesecke, G., Theil, F.: Validity and failure of the cauchy-born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12, 445–478 (2002)
Garcke, H.: On mathematical models for phase separation in elastically stressed solids. Habilitation thesis, University Bonn (2001)
Gurtin, M.E., Polignone, D., Viñals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Model. and Math. Sci. 2, 191–211 (1996)
Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98, 123–142 (1987)
Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University Press, New Jersey (1997)
Sternberg, P.: Vector-valued local minimisers of nonconvex variational problems. Rocky Mountain J. Math. 21, 799–807 (1991)
Weikard, U.: Numerische Lösungen der Cahn–Hilliard–Gleichung unter Einbeziehung elastischer Materialeigenschaften. Diploma thesis, University Bonn (1998)
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Communicated by A. Zhou.
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Blesgen, T. On the competition of elastic energy and surface energy in discrete numerical schemes. Adv Comput Math 27, 179–194 (2007). https://doi.org/10.1007/s10444-007-9028-5
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DOI: https://doi.org/10.1007/s10444-007-9028-5