Home A Parametric Finite Element Method for a Degenerate Multi-Phase Stefan Problem with Triple Junctions
Article
Licensed
Unlicensed Requires Authentication

A Parametric Finite Element Method for a Degenerate Multi-Phase Stefan Problem with Triple Junctions

  • Tokuhiro Eto ORCID logo EMAIL logo , Harald Garcke ORCID logo and Robert Nürnberg ORCID logo
Published/Copyright: October 22, 2025

Abstract

In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

MSC 2020: 35R35; 65M12; 65M50; 65M60; 80A22

References

[1] H. Abels, M. Rauchecker and M. Wilke, Well-posedness and qualitative behaviour of the Mullins–Sekerka problem with ninety-degree angle boundary contact, Math. Ann. 381 (2021), no. 1–2, 363–403. 10.1007/s00208-020-02007-3Search in Google Scholar

[2] E. Bänsch, K. Deckelnick, H. Garcke and P. Pozzi, Interfaces: Modeling, Analysis, Numerics, Oberwolfach Semin. 51, Birkhäuser/Springer, Cham 2023. 10.1007/978-3-031-35550-9Search in Google Scholar

[3] W. Bao, H. Garcke, R. Nürnberg and Q. Zhao, A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters, Numer. Methods Partial Differential Equations 39 (2023), no. 1, 759–794. 10.1002/num.22921Search in Google Scholar

[4] W. Bao and Y. Li, A symmetrized parametric finite element method for anisotropic surface diffusion in three dimensions, SIAM J. Sci. Comput. 45 (2023), no. 4, A1438–A1461. 10.1137/22M1500575Search in Google Scholar

[5] W. Bao and Y. Li, A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy, Numer. Math. 156 (2024), no. 2, 609–639. 10.1007/s00211-024-01398-8Search in Google Scholar

[6] W. Bao and Q. Zhao, A structure-preserving parametric finite element method for surface diffusion, SIAM J. Numer. Anal. 59 (2021), no. 5, 2775–2799. 10.1137/21M1406751Search in Google Scholar

[7] J. W. Barrett and J. F. Blowey, An error bound for the finite element approximation of the Cahn–Hilliard equation with logarithmic free energy, Numer. Math. 72 (1995), no. 1, 1–20. 10.1007/s002110050157Search in Google Scholar

[8] J. W. Barrett, J. F. Blowey and H. Garcke, Finite element approximation of the Cahn–Hilliard equation with degenerate mobility, SIAM J. Numer. Anal. 37 (1999), no. 1, 286–318. 10.1137/S0036142997331669Search in Google Scholar

[9] J. W. Barrett, J. F. Blowey and H. Garcke, On fully practical finite element approximations of degenerate Cahn–Hilliard systems, M2AN Math. Model. Numer. Anal. 35 (2001), no. 4, 713–748. 10.1051/m2an:2001133Search in Google Scholar

[10] J. W. Barrett, H. Garcke and R. Nürnberg, A parametric finite element method for fourth order geometric evolution equations, J. Comput. Phys. 222 (2007), no. 1, 441–462. 10.1016/j.jcp.2006.07.026Search in Google Scholar

[11] J. W. Barrett, H. Garcke and R. Nürnberg, On the variational approximation of combined second and fourth order geometric evolution equations, SIAM J. Sci. Comput. 29 (2007), no. 3, 1006–1041. 10.1137/060653974Search in Google Scholar

[12] J. W. Barrett, H. Garcke and R. Nürnberg, Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering, European J. Appl. Math. 21 (2010), no. 6, 519–556. 10.1017/S0956792510000203Search in Google Scholar

[13] J. W. Barrett, H. Garcke and R. Nürnberg, On stable parametric finite element methods for the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth, J. Comput. Phys. 229 (2010), no. 18, 6270–6299. 10.1016/j.jcp.2010.04.039Search in Google Scholar

[14] J. W. Barrett, H. Garcke and R. Nürnberg, Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies, Interfaces Free Bound. 12 (2010), no. 2, 187–234. 10.4171/ifb/232Search in Google Scholar

[15] J. W. Barrett, H. Garcke and R. Nürnberg, Parametric finite element approximations of curvature-driven interface evolutions, Geometric Partial Differential Equations. Part I, Handb. Numer. Anal. 21, Elsevier/North-Holland, Amsterdam (2020), 275–423. 10.1016/bs.hna.2019.05.002Search in Google Scholar

[16] P. W. Bates and S. Brown, A numerical scheme for the Mullins–Sekerka evolution in three space dimensions, Differential Equations and Computational Simulations (Chengdu 1999), World Scientific, River Edge (2000), 12–26. Search in Google Scholar

[17] P. W. Bates, X. Chen and X. Deng, A numerical scheme for the two phase Mullins–Sekerka problem, Electron. J. Differential Equations 1995 (1995), 1–27. Search in Google Scholar

[18] J. F. Blowey, M. I. M. Copetti and C. M. Elliott, Numerical analysis of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal. 16 (1996), no. 1, 111–139. 10.1093/imanum/16.1.111Search in Google Scholar

[19] J. F. Blowey and C. M. Elliott, The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis, European J. Appl. Math. 3 (1992), no. 2, 147–179. 10.1017/S0956792500000759Search in Google Scholar

[20] E. Bretin, R. Denis, S. Masnou, A. Sengers and G. Terii, A multiphase Cahn–Hilliard system with mobilities and the numerical simulation of dewetting, ESAIM Math. Model. Numer. Anal. 57 (2023), no. 3, 1473–1509. 10.1051/m2an/2023023Search in Google Scholar

[21] L. Bronsard, H. Garcke and B. Stoth, A multi-phase Mullins–Sekerka system: Matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 3, 481–506. 10.1017/S0308210500021612Search in Google Scholar

[22] A. Chambolle and T. Laux, Mullins–Sekerka as the Wasserstein flow of the perimeter, Proc. Amer. Math. Soc. 149 (2021), no. 7, 2943–2956. 10.1090/proc/15401Search in Google Scholar

[23] C. Chen, C. Kublik and R. Tsai, An implicit boundary integral method for interfaces evolving by Mullins–Sekerka dynamics, Mathematics for Nonlinear Phenomena—Analysis and Computation, Springer Proc. Math. Stat. 215, Springer, Cham (2017), 1–21. 10.1007/978-3-319-66764-5_1Search in Google Scholar

[24] S. Chen, B. Merriman, S. Osher and P. Smereka, A simple level set method for solving Stefan problems, J. Comput. Phys. 135 (1997), no. 1, 8–29. 10.1006/jcph.1997.5721Search in Google Scholar

[25] X. Chen, J. Hong and F. Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins–Sekerka problem, Comm. Partial Differential Equations 21 (1996), no. 11–12, 1705–1727. 10.1080/03605309608821243Search in Google Scholar

[26] T. A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software 30 (2004), no. 2, 196–199. 10.1145/992200.992206Search in Google Scholar

[27] T. A. Davis, Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization, ACM Trans. Math. Software 38 (2011), no. 1, 1–22. 10.1145/2049662.2049670Search in Google Scholar

[28] C. M. Elliott and D. A. French, Numerical studies of the Cahn–Hilliard equation for phase separation, IMA J. Appl. Math. 38 (1987), no. 2, 97–128. 10.1093/imamat/38.2.97Search in Google Scholar

[29] J. Escher, A.-V. Matioc and B.-V. Matioc, The Mullins–Sekerka problem via the method of potentials, Math. Nachr. 297 (2024), no. 5, 1960–1977. 10.1002/mana.202300350Search in Google Scholar

[30] J. Escher and G. Simonett, Classical solutions for Hele–Shaw models with surface tension, Adv. Differential Equations 2 (1997), no. 4, 619–642. 10.57262/ade/1366741151Search in Google Scholar

[31] J. Escher and G. Simonett, A center manifold analysis for the Mullins–Sekerka model, J. Differential Equations 143 (1998), no. 2, 267–292. 10.1006/jdeq.1997.3373Search in Google Scholar

[32] T. Eto, A rapid numerical method for the Mullins–Sekerka flow with application to contact angle problems, J. Sci. Comput. 98 (2024), no. 3, Paper No. 63. 10.1007/s10915-024-02469-6Search in Google Scholar

[33] T. Eto, H. Garcke and R. Nürnberg, A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions, Numer. Math. 156 (2024), no. 4, 1479–1509. 10.1007/s00211-024-01414-xSearch in Google Scholar

[34] D. J. Eyre, Systems of Cahn–Hilliard equations, SIAM J. Appl. Math. 53 (1993), no. 6, 1686–1712. 10.1137/0153078Search in Google Scholar

[35] X. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn–Hilliard equation, Numer. Math. 99 (2004), no. 1, 47–84. 10.1007/s00211-004-0546-5Search in Google Scholar

[36] X. Feng and A. Prohl, Numerical analysis of the Cahn–Hilliard equation and approximation of the Hele–Shaw problem, Interfaces Free Bound. 7 (2005), no. 1, 1–28. 10.4171/ifb/111Search in Google Scholar

[37] J. Fischer, S. Hensel, T. Laux and T. M. Simon, A weak-strong uniqueness principle for the Mullins–Sekerka equation, preprint (2024), https://arxiv.org/abs/2404.02682. Search in Google Scholar

[38] H. Garcke, Curvature driven interface evolution, Jahresber. Dtsch. Math.-Ver. 115 (2013), no. 2, 63–100. 10.1365/s13291-013-0066-2Search in Google Scholar

[39] H. Garcke, R. Nürnberg and Q. Zhao, Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches, J. Comput. Phys. 489 (2023), Article ID 112276. 10.1016/j.jcp.2023.112276Search in Google Scholar

[40] H. Garcke, R. Nürnberg and Q. Zhao, A variational front-tracking method for multiphase flow with triple junctions, Math. Comp. (2025), 10.1090/mcom/4078. 10.1090/mcom/4078Search in Google Scholar

[41] H. Garcke and M. Rauchecker, Stability analysis for stationary solutions of the Mullins–Sekerka flow with boundary contact, Math. Nachr. 295 (2022), no. 4, 683–705. 10.1002/mana.201900303Search in Google Scholar

[42] H. Garcke and T. Sturzenhecker, The degenerate multi-phase Stefan problem with Gibbs–Thomson law, Adv. Math. Sci. Appl. 8 (1998), no. 2, 929–941. Search in Google Scholar

[43] S. Hensel and K. Stinson, Weak solutions of Mullins–Sekerka flow as a Hilbert space gradient flow, Arch. Ration. Mech. Anal. 248 (2024), no. 1, Paper No. 8. 10.1007/s00205-023-01950-0Search in Google Scholar

[44] F. Izsák and T.-E. Djebbar, Learning data for neural-network-based numerical solution of PDEs: Application to Dirichlet-to-Neumann problems, Algorithms 16 (2023), 10.3390/a16020111. 10.3390/a16020111Search in Google Scholar

[45] M. Li and Q. Zhao, Parametric finite element approximations for anisotropic surface diffusion with axisymmetric geometry, J. Comput. Phys. 497 (2024), Article ID 112632. 10.1016/j.jcp.2023.112632Search in Google Scholar

[46] Y. Li, J.-I. Choi and J. Kim, Multi-component Cahn–Hilliard system with different boundary conditions in complex domains, J. Comput. Phys. 323 (2016), 1–16. 10.1016/j.jcp.2016.07.017Search in Google Scholar

[47] Y. Li, R. Liu, Q. Xia, C. He and Z. Li, First- and second-order unconditionally stable direct discretization methods for multi-component Cahn–Hilliard system on surfaces, J. Comput. Appl. Math. 401 (2022), Article ID 113778. 10.1016/j.cam.2021.113778Search in Google Scholar

[48] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 253–271. 10.1007/BF01205007Search in Google Scholar

[49] U. F. Mayer, A numerical scheme for moving boundary problems that are gradient flows for the area functional, European J. Appl. Math. 11 (2000), no. 1, 61–80. 10.1017/S0956792599003812Search in Google Scholar

[50] R. Nürnberg, Numerical simulations of immiscible fluid clusters, Appl. Numer. Math. 59 (2009), no. 7, 1612–1628. 10.1016/j.apnum.2008.11.003Search in Google Scholar

[51] R. Nürnberg, A structure preserving front tracking finite element method for the Mullins–Sekerka problem, J. Numer. Math. 31 (2023), no. 2, 137–155. 10.1515/jnma-2021-0131Search in Google Scholar

[52] M. Röger, Existence of weak solutions for the Mullins–Sekerka flow, SIAM J. Math. Anal. 37 (2005), no. 1, 291–301. 10.1137/S0036141004439647Search in Google Scholar

[53] A. Schmidt and K. G. Siebert, Design of Adaptive Finite Element Software, Lect. Notes Comput. Sci. Eng. 42, Springer, Berlin, 2005. Search in Google Scholar

[54] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1427–1451. 10.3934/dcds.2011.31.1427Search in Google Scholar

[55] J. Zhu, X. Chen and T. Y. Hou, An efficient boundary integral method for the Mullins–Sekerka problem, J. Comput. Phys. 127 (1996), no. 2, 246–267. 10.1006/jcph.1996.0173Search in Google Scholar

Received: 2025-05-20
Revised: 2025-08-20
Accepted: 2025-09-28
Published Online: 2025-10-22

© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston

Downloaded on 25.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/cmam-2025-0080/html?lang=en
Scroll to top button