EDP Sciences logo
Subscriber Authentication Point
Sciengine logo
Search
Menu
All issues Volume 60 / No 3 (May-June 2026) ESAIM: M2AN, 60 3 (2026) 1363-1410 References
Open Access
Issue
ESAIM: M2AN
Volume 60, Number 3, May-June 2026
Page(s) 1363 - 1410
DOI https://doi.org/10.1051/m2an/2026035
Published online 05 June 2026
  1. J.P. Ahrens, B. Geveci and C.C. Law, Paraview: An End-User Tool for Large-Data Visualization. The Visualization Handbook (2005). [Google Scholar]
  2. S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. [Google Scholar]
  3. J.W. Barrett and J.F. Blowey, An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 19 (1999) 147–168. [Google Scholar]
  4. J.W. Barrett and J.F. Blowey, An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. ESAIM: Math. Modell. Numer. Anal. 33 (1999) 971–987. [Google Scholar]
  5. J.W. Barrett, J.F. Blowey and H. Garcke, On fully practical finite element approximations of degenerate Cahn–Hilliard systems. ESAIM: Math. Modell. Numer. Anal. 35 (2001) 713–748. [Google Scholar]
  6. F. Boyer and C. Lapuerta, Study of a three component Cahn–Hilliard flow model. ESAIM: Math. Modell. Numer. Anal. 40 (2006) 653–687. [Google Scholar]
  7. F. Boyer and S. Minjeaud, Numerical schemes for a three component Cahn–Hilliard model. ESAIM: Math. Modell. Numer. Anal. 45 (2011) 697–738. [Google Scholar]
  8. F. Boyer and S. Minjeaud, Hierarchy of consistent n-component Cahn–Hilliard systems. Math. Models Methods Appl. Sci. 24 (2014) 2885–2928. [Google Scholar]
  9. J. Brannick, C. Liu, T. Qian and H. Sun, Diffuse interface methods for multiple phase materials: an energetic variational approach. Numer. Math.: Theory Methods App. 8 (2015) 220–236. [Google Scholar]
  10. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3 edition. Springer New York, NY (2008). [Google Scholar]
  11. R.C. Cabrales, F. Guill´en-González and J.V. Guti´errez-Santacreu, A time-splitting finite-element stable approximation for the Ericksen–Leslie equations. SIAM J. Sci. Comput. 37 (2015) B261–B282. [Google Scholar]
  12. L. Caffarelli and N. Muler, An l∞ bound for solutions of the Cahn–Hilliard equation. Arch. Ration. Mech. Anal. 133 (1995) 129–144. [Google Scholar]
  13. J.W. Cahn, On spinodal decomposition. Acta Metall. 9 (2011) 795–801. [Google Scholar]
  14. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy.J. Chem. Phys. 28 (1958) 258–267. [Google Scholar]
  15. W. Chen, C. Wang, S. Wang, X. Wang and S.M. Wise, Energy stable numerical schemes for ternary Cahn–Hilliard system. J. Sci. Comput. 84 (2020) 27. [Google Scholar]
  16. Q. Cheng, X. Yang and J. Shen, Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. J. Comput. Phys. 341 (2017) 44–60. [Google Scholar]
  17. S. Dong, An efficient algorithm for incompressible n-phase flows. J. Comput. Phys. 276 (2014) 691–728. [Google Scholar]
  18. S. Dong, Physical formulation and numerical algorithm for simulating n immiscible incompressible fluids involving general order parameters. J. Comput. Phys. 283 (2015) 98–128. [Google Scholar]
  19. M. El Haddad and G. Tierra, A thermodynamically consistent model for two-phase incompressible flows with different densities. Derivation and efficient energy-stable numerical schemes. Comput. Methods Appl. Mech. Eng. 389 (2022) 114328. [Google Scholar]
  20. C.M. Elliott and S. Luckhaus, A Generalised Diffusion Equation for Phase Separation of a Multi-Component Mixture with Interfacial Free Energy. University of Minnesota. Institute for Mathematics and Its Applications (1991). [Google Scholar]
  21. A. Ern and J.-L. Guermond, Finite Elements I: Approximation and Interpolation. Texts in Applied Mathematics. Springer International Publishing (2021). [Google Scholar]
  22. D.J. Eyre, Systems of cahn-hilliard equations. SIAM J. Appl. Math. 53 (1993) 1686–1712. [Google Scholar]
  23. V. Girault and P. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986). [Google Scholar]
  24. Y. Gong, J. Zhao and Q. Wang, Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models. Comput. Phys. Commun. 249 (2020) 107033. [Google Scholar]
  25. Y. Gong, J. Zhao and Q. Wang, Arbitrarily high-order unconditionally energy stable schemes for thermodynamically consistent gradient flow models. SIAM J. Sci. Comput. 42 (2020) B135–B156. [Google Scholar]
  26. F. Guill´en-González and G. Tierra, On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234 (2013) 140–171. [CrossRef] [MathSciNet] [Google Scholar]
  27. F. Guill´en-González and G. Tierra, Second order schemes and time-step adaptivity for Allen–Cahn and Cahn–Hilliard models. Comput. Math. App. 68 (2014) 821–846. [Google Scholar]
  28. F. Guill´en-González and G. Tierra, Unconditionally energy stable numerical schemes for phase-field vesicle membrane model. J. Comput. Phys. 354 (2018) 67–85. [Google Scholar]
  29. F. Hecht, New development in FreeFem++. J. Num. Math. 20 (2012) 1–14. [Google Scholar]
  30. P.C. Hohenberg and B.I. Halperin, Theory of dynamic critical phenomena. Rev. Mod. Phys. 49 (1977) 435–479. [CrossRef] [Google Scholar]
  31. T.M. Inc., Matlab version: 9.13.0 (r2022b) (2022). [Google Scholar]
  32. J. Kim, Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12 (2012) 613–661. [Google Scholar]
  33. J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193 (2004) 511–543. [CrossRef] [MathSciNet] [Google Scholar]
  34. H.G. Lee, J. Shin and J.-Y. Lee, A high-order unconditionally energystable scheme for the ternary Cahn–Hilliard system (2024). [Google Scholar]
  35. S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model. Numer. Methods Part. Differ. Equ. 29 (2013) 584–618. [CrossRef] [Google Scholar]
  36. R. Nochetto, A. Salgado and S. Walker, A diffuse interface model for electrowetting with moving contact lines. Math. Models Methods Appl. Sci. 24 (2014) 67–111. [Google Scholar]
  37. R. Nurnberg, Numerical simulations of immiscible fluid clusters. Appl. Numer. Math. 29 (2009) 1612–1628. [Google Scholar]
  38. J. Shen and X. Yang, Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691. [CrossRef] [MathSciNet] [Google Scholar]
  39. J. Swain and G. Tierra, Linear numerical schemes for a q-tensor system for nematic liquid crystals. Comput. Methods Appl. Mech. Eng. 430 (2024) 117190. [Google Scholar]
  40. S. Wu and J. Xu, Multiphase Allen–Cahn and Cahn–Hilliard models and their discretizations with the effect of pairwise surface tensions. J. Comput. Phys. 343 (2017) 10–32. [Google Scholar]
  41. X. Wu, G. van Zwieten and K.G. van der Zee, Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed. Eng. 30 (2014) 180–203. [Google Scholar]
  42. J. Wu, J. Yang and Z. Tan, Highly efficient variant of SAV approach for the incompressible multi-component phasefield fluid models. Comput. Math. App. 145 (2023) 24–40. [Google Scholar]
  43. J. Xu, Y. Li, S. Wu and A. Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Comput. Methods Appl. Mech. Eng. 345 (2019) 826–853. [Google Scholar]
  44. J. Xu, G. Vilanova and H. Gomez, Phase-field model of vascular tumor growth: three-dimensional geometry of the vascular network and integration with imaging data. Comput. Meth. Appl. Mech. Eng. 359 (2020) 112648. [Google Scholar]
  45. Z. Yang and S. Dong, Multiphase flows of n immiscible incompressible fluids: an outflow/open boundary condition and algorithm. J. Comput. Phys. 366 (2018) 33–70. [Google Scholar]
  46. J. Yang and J. Kim, An unconditionally stable second-order accurate method for systems of Cahn–Hilliard equations. Commun. Nonlinear Sci. Numer. Simul. 87 (2020) 105276. [CrossRef] [MathSciNet] [Google Scholar]
  47. J. Yang and J. Kim, Linear, second-order accurate, and energy stable scheme for a ternary Cahn–Hilliard model by using Lagrange multiplier approach. Acta Appl. Math. 172 (2021) 1–20. [Google Scholar]
  48. X. Yang, J. Zhao, Q.Wang and J. Shen, Numerical approximations for a three-components Cahn–Hilliard phase-field model based on the invariant energy quadratization method. Math. Models Methods Appl. Sci. 27 (2017) 1993–2030. [Google Scholar]
  49. M. Yuan, W. Chen, C. Wang, S.M. Wise and Z. Zhang, An energy stable finite element scheme for the threecomponent Cahn–Hilliard-type model for macromolecular microsphere composite hydrogels. J. Sci. Comput. 87 (2021) 78. [Google Scholar]
  50. J. Zhang and X. Yang, Decoupled, non-iterative, and unconditionally energy stable large time stepping method for the three-phase Cahn–Hilliard phase-field model. J. Comput. Phys. 404 (2020) 109115. [Google Scholar]
  51. J. Zhao, H. Li, Q. Wang and X. Yang, Decoupled energy stable schemes for a phase field model of three-phase incompressible viscous fluid flow. J. Sci. Comput. 70 (2017) 1367–1389. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you