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Analysis of a time discretization scheme for a nonstandardviscous Cahn–Hilliard system

Published online by Cambridge University Press:  30 June 2014

Pierluigi Colli ,
Gianni Gilardi ,
Pavel Krejčí ,
Paolo Podio-Guidugli  and
Jürgen Sprekels
Pierluigi Colli
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy.. pierluigi.colli@unipv.it; gianni.gilardi@unipv.it
Gianni Gilardi
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy.. pierluigi.colli@unipv.it; gianni.gilardi@unipv.it
Pavel Krejčí
Affiliation:
Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic.; krejci@math.cas.cz
Paolo Podio-Guidugli
Affiliation:
Accademia Nazionale dei Lincei and Department of Mathematics, University of Rome TorVergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy.; ppg@uniroma2.it
Jürgen Sprekels
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany. ; sprekels@wias-berlin.de
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Abstract

In this paper we propose a time discretization of a system of two parabolic equationsdescribing diffusion-driven atom rearrangement in crystalline matter. The equationsexpress the balances of microforces and microenergy; the two phase fields are the orderparameter and the chemical potential. The initial and boundary-value problem for theevolutionary system is known to be well posed. Convergence of the discrete scheme to thesolution of the continuous problem is proved by a careful development of uniformestimates, by weak compactness and a suitable treatment of nonlinearities. Moreover, forthe difference of discrete and continuous solutions we prove an error estimate of orderone with respect to the time step.

Keywords

Cahn–Hilliard equationphase field modeltime discretizationconvergenceerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1061 - 1087
Copyright
© EDP Sciences, SMAI 2014

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References

Bai, F., Elliott, C.M., Gardiner, A., Spence, A. and Stuart, A.M., The viscous Cahn-Hilliard equation. I. Computations. Nonlinearity 8 (1995) 131–160. Google Scholar
Barrett, J.W. and Blowey, J.F., An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Numer. Math. 72 (1995) 1–20. Google Scholar
Barrett, J.W. and Blowey, J.F., An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix. Numer. Math. 88 (2001) 255–297. Google Scholar
Barrett, J.W., Blowey, J.F. and Garcke, H., Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. Google Scholar
Barrett, J.W., Blowey, J.F. and Garcke, H., On fully practical finite element approximations of degenerate Cahn-Hilliard systems. ESAIM: M2AN 35 (2001) 713–748. Google Scholar
Bartels, S. and Müller, R., A posteriori error controlled local resolution of evolving interfaces for generalized Cahn-Hilliard equations. Interfaces Free Bound. 12 (2010) 45–73. Google Scholar
Bartels, S. and Müller, R., Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential. Numer. Math. 119 (2011) 409–435. Google Scholar
Blowey, J.F. and Elliott, C.M., The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3 (1992) 147–179. Google Scholar
Bonetti, E., Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence. Quart. Appl. Math. 62 (2004) 53–76. Google Scholar
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, no. 5. Notas de Matemática. North-Holland Publishing Co., Amsterdam (1973).
Carstensen, C. and Plecháč, P., Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. ESAIM: M2AN 35 (2001) 865–878. Google Scholar
Chen, Z., Nochetto, R.H. and Schmidt, A., Error control and adaptivity for a phase relaxation model. ESAIM: M2AN 34 (2000) 775–797. Google Scholar
Cherfils, L., Petcu, M. and Pierre, M., A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 27 (2010) 1511–1533. Google Scholar
Chiodaroli, E., A dissipative model for hydrogen storage: existence and regularity results. Math. Methods Appl. Sci. 34 (2011) 642–669. Google Scholar
Colli, P., Frémond, M. and Klein, O., Global existence of a solution to a phase field model for supercooling. Nonlinear Anal. Real World Appl. 2 (2001) 523–539. Google Scholar
Colli, P., Gilardi, G., Podio-Guidugli, P. and Sprekels, J., Well-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system. SIAM J. Appl. Math. 71 (2011) 1849–1870. Google Scholar
Colli, P., Gilardi, G., Podio-Guidugli, P. and Sprekels, J., Global existence and uniqueness for a singular/degenerate Cahn–Hilliard system with viscosity. J. Differ. Equ. 254 (2013) 4217–4244. Google Scholar
Eck, C., Jadamba, B. and Knabner, P., Error estimates for a finite element discretization of a phase field model for mixtures. SIAM J. Numer. Anal. 47 (2010) 4429–4445. Google Scholar
Frigeri, S., Krejčí, P. and Stefanelli, U., Quasistatic isothermal evolution of shape memory alloys. Math. Models Methods Appl. Sci. 21 (2011) 2409–2432. Google Scholar
Gilardi, G. and Stefanelli, U., Time-discretization and global solution for a doubly nonlinear Volterra equation. J. Differ. Equ. 228 (2006) 707–736. Google Scholar
Gräser, C. and Kornhuber, R., Multigrid methods for obstacle problems. J. Comput. Math. 27 (2009) 1–44. Google Scholar
Gräser, C. and Kornhuber, R., Nonsmooth Newton methods for set-valued saddle point problems. SIAM J. Numer. Anal. 47 (2009) 1251–1273. Google Scholar
C. Gräser, R. Kornhuber and U. Sack, Nonsmooth Schur-Newton methods for vector-valued Cahn-Hilliard equations. Freie Universität Berlin, Fachbereich Mathematik und Informatik, Serie A Preprint no. 01 (2013) 1–16.
J.W. Jerome, Approximation of nonlinear evolution systems, vol. 164 of Math. Sci. Eng. Academic Press Inc., Orlando, FL (1983).
Kessler, D. and Scheid, J.-F., A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy. IMA J. Numer. Anal. 22 (2002) 281–305. Google Scholar
Kim, J., Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (2012) 613–661. Google Scholar
Krejčí, P. and Stefanelli, U., Well-posedness of a thermo-mechanical model for shape memory alloys under tension. ESAIM: M2AN 44 (2010) 1239–1253. Google Scholar
Mielke, A., Paoli, L. and Petrov, A., On existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys. SIAM J. Math. Anal. 41 (2009) 1388–1414. Google Scholar
Podio-Guidugli, P., Models of phase segregation and diffusion of atomic species on a lattice. Ric. Mat. 55 (2006) 105–118. Google Scholar
Röger, M., Existence of weak solutions for the Mullins-Sekerka flow. SIAM J. Math. Anal. 37 (2005) 291–301. Google Scholar
Segatti, A., Error estimates for a variable time-step discretization of a phase transition model with hyperbolic momentum. Numer. Funct. Anal. Optim. 25 (2004) 547–569. Google Scholar
Simon, J., Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. Google Scholar
Stefanelli, U., Error control of a nonlinear evolution problem related to phase transitions. Numer. Funct. Anal. Optim. 20 (1999) 585–608. Google Scholar
Stefanelli, U., Error control for a time-discretization of the full one-dimensional Frémond model for shape memory alloys. Adv. Math. Sci. Appl. 10 (2000) 917–936. Google Scholar
Stefanelli, U., Analysis of a variable time-step discretization for a phase transition model with micro-movements. Commun. Pure Appl. Anal. 5 (2006) 657–671. Google Scholar
Vaz, C.L.D., Rothe’s method for an isothermal phase-field model of a binary alloy with convection. Mat. Contemp. 32 (2007) 221–251. Google Scholar