Skip to main content
SpringerLink
Log in

A Splitting Method for the Allen-Cahn/Cahn-Hilliard System Coupled with Heat Equation Based on Maxwell-Cattaneo Law

  • Original Paper
  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

We analyze here a finite element space semidiscretization of the Allen-Cahn/Cahn-Hilliard system coupled with heat equation and based on the Maxwell-Cattaneo law. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116, 5–16 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barrett, J., Blowey, J.: Finite element approximation of a degenerate Allen-Cahn/Cahn-Hilliard system. SIAM J. Nume. Anal. 39(5), 1598–1624 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brochet, D., Hilhorst, D., Novick-Cohen, A.: Finite-dimensional exponential attractor for a model for order-disorder and phase separation. Appl. Math. Lett. 7, 83–87 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cahn, J.W., Novick-Cohen, A.: Evolution equations for phase separation and ordering in binary alloys. Statistical Phys. 76, 877–909 (1994)

    Article  MATH  Google Scholar 

  5. Cherfils, L., Petcu, M.: A numerical analysis of the Cahn-Hilliard equation with non-permeable walls. Numer. Math. 128, 518–549 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cherfils, L., Petcu, M., Pierre, M.: A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete Contin. Dyn. Syst. 8, 1511–1533 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christov, C.I., Jordan, P.M.: Heat conduction paradox involving second-sound propagation in moving media. Phys. Rev. Lett. 94, 154301 (2005)

    Article  Google Scholar 

  8. Ciarlet, P.G.: The finite element method for elliptic problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2002)

  9. Del Passo, R., Giacomelli, L., Novick-Cohen, A.: Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility. Interfaces free boundaries 1, 199–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derkach, V., Novick-Cohen, A., Vilenkin, A.: Geometric interfacial motion: Coupling surface diffusion and mean curvature motion, submitted

  11. El-Azab, A., Ahmed, K., Rokkam, S., Hochrainer, T.: Diffuse interface modeling of void growth in irradiated materials. Mathematical, thermodynamic and atomistic perspectives, COSSMS 18, 90–98 (2014)

    Google Scholar 

  12. Elliott, C.M., French, D.A.: A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ern, A., Guermond, J.-L.: Éléments finis: Théorie, applications, mise en oeuvre. Springer, Berlin (2002)

    MATH  Google Scholar 

  14. Goldstein, G.R., Miranville, A., Schimperna, G.: A Cahn-Hilliard equation in a domain with non-permeable walls. Phys. D. 240(8), 754–766 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grasselli, M., Pierre, M.: A splitting method for the Cahn-Hilliard equation with inertial term. Math. Models Methods Appl. Sci. 20(8), 1363–1390 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grasselli, M., Pierre, M.: Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Commun. Pure Appl. Anal. 11, 2393–2416 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grasselli, M., Petzeltová, H., Schimperna, G.: Long time behavior of solutions to the Caginalp system with singular potential. Zeitschrift für Analysis und ihre Anwendungen 25(1), 51–72 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Haraux, A., Jendoubi, M.A.: Convergence of solutions of second-order gradient-like systems with analytic nonlinearities. J. Diff. Eqns. 144(2), 313–320 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, J.: Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law. J. Math. Anal. Appl. 341, 149–169 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, J.: Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law. Math. Methods Appl. Sci. 32, 1156–1182 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lojasiewicz, S.: Ensembles semi-analytiques, I.H.E.S. Notes (1965)

  22. Makki, A., Miranville, A., Saoud, W.: On a Cahn-Hilliard/Allen-Cahn system coupled with a type \(III\) heat equation and singular potentials, Nonlinear Anal. 196, (2020)

  23. Makki, A., Miranville, A., Sadaka, G.: On the nonconserved Caginalp phase-field system based on the maxwell Cattaneo law with two temperatures and logarithmic potentials. Discrete Contin. Dyn. Systems Ser. B 24, 1341–1365 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Makki, A., Miranville, A., Sadaka, G.: On the conserved Caginalp phase-field system logarithmic potentials based on the maxwell-Cattaneo law with two temperatures. Appl Math Optim 84, 1285–1316 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Makki, A., Miranville, A., Petcu, M.: A numerical analysis of the coupled Cahn-Hilliard/Allen-Cahn system with dynamic boundary conditions. Int. J. Numer. Anal. Mod. 19, 630–655 (2022)

    MathSciNet  MATH  Google Scholar 

  26. Merlet, B., Pierre, M.: Convergence to equilibrium for the Euler backward scheme and applications. Commun. Pure Appl. Anal. 9(3), 685–702 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Millett, P.C., Rokkam, S., El-Azab, A., Tonks, M., Wolf, D.: Void nucleation and growth in irradiated polycrystalline metals: A phase-field model. Modelling Simul. Mater. Sci. Eng. 17, 0064003 (2009)

    Article  Google Scholar 

  28. Miranville, A.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28, 709–735 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Miranville, A., Quintanilla, R.: A generalization of the Caginalp phase-field system based on the Cattaneo heat flux law. Nonlinear Anal. TMA 71, 2278–2290 (2009)

    Article  MATH  Google Scholar 

  30. Miranville, A., Quintanilla, R.: Some generalizations of the Caginalp phase-field system. Appl. Anal. 88, 877–894 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Miranville, A., Quintanilla, R.: A phase-field model based on a three-phase-lag heat conduction. Appl. Math. Optim. 63, 133–150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Miranville, A., Quintanilla, R.: A type III phase-field system with a logarithmic potential. Appl. Math. Letters 24, 1003–1008 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Miranville, A., Quintanilla, R.: On a phase-field system based on the Cattaneo law. Nonlinear Anal. TMA 75, 2552–2565 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Miranville, A., Quintanilla, R.: On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law. Math. Methods Appl. Sci. 35, 4385–4397 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Miranville, A., Saoud, W., Talhouk, R.: Asymptotic behavior of a model for order-disorder and phase separation. Asympt. Anal. 103, 57–76 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Miranville, A., Saoud, W., Talhouk, R.: On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete Contin. Dyn. Syst. Ser. B 24, 2278–2290 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Miranville, A., Quintanilla, R., Saoud, W.: Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Comm. Pure Appl. Anal. 19(4), 2257–2288 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  38. Novick-Cohen, A.: Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system. Phys. D 137, 1–24 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Novick-Cohen, A., Peres Hari, L.: Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case. Phys. D 209, 205–235 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rokkam, S., El-Azab, A., Millett, P., Wolf, D.: Phase field modeling of void nucleation and growth in irradiated metals. Modelling Simul. Mater. Sci. Eng. 17, 0064002 (2009)

    Article  Google Scholar 

  41. Temam, R.: Infinite-Dimensional dynamical systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)

    Book  MATH  Google Scholar 

  42. Thomée, V.: Galerkin finite element methods for parabolic problems, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  43. Tonks, M.R., Gaston, D., Millett, P.C., Andrs, D., Talbot, P.: An object-oriented finite element framework for multiphysics phase field simulations. Comput. Mater. Sci. 51, 20–29 (2012)

    Article  Google Scholar 

  44. Wang, L., Lee, J., Anitescu, M., Azab, A.E., Mcinnes, L.C., Munson, T., Smith, B.: A differential variational inequality approach for the simulation of heterogeneous materials, in: Proc. SciDAC, (2011)

  45. Xia, Y., Xu, Y., Shu, C.W.: Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. Commun. Comput. Phys. 5, 821–835 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Yang, C., Cai, X.C., Keyes, D.E., Pernice, M.: NKS Method for the Implicit Solution of a coupled Allen-Cahn/Cahn-Hilliard System, Proceedings of the 21th International Conference on Domain Decomposition Methods, (2012)

  47. Yang, C., Liu, C.: A further study on the coupled Allen-Cahn/Cahn-Hilliard equations, Bound. value Probl. 54, (2019)

Download references

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and many useful comments which helped to improve it.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Departement of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Nader El Khatib & Ahmad Makki

  2. Laboratoire de Mathématiques et Applications, Université de Poitiers, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962, Chasseneuil Futuroscope Cedex, France

    Madalina Petcu

  3. The Institute of Mathematics of the Romanian Academy, Bucharest, Romania

    Madalina Petcu

Authors
  1. Nader El Khatib
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmad Makki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Madalina Petcu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmad Makki.

Ethics declarations

Competing Interests

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Khatib, N., Makki, A. & Petcu, M. A Splitting Method for the Allen-Cahn/Cahn-Hilliard System Coupled with Heat Equation Based on Maxwell-Cattaneo Law. Appl Math Optim 88, 17 (2023). https://doi.org/10.1007/s00245-023-09990-4

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-023-09990-4

Keywords

Mathematics Subject Classification

Search

Navigation