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A Scalar Auxiliary Variable Unfitted FEM for the Surface Cahn–Hilliard Equation

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Abstract

The paper studies a scalar auxiliary variable (SAV) method to solve the Cahn–Hilliard equation with degenerate mobility posed on a smooth closed surface \(\Gamma \). The SAV formulation is combined with adaptive time stepping and a geometrically unfitted trace finite element method (TraceFEM), which embeds \(\Gamma \) in \(\mathbb {R}^3\). The stability is proven to hold in an appropriate sense for both first- and second-order in time variants of the method. The performance of our SAV method is illustrated through a series of numerical experiments, which include systematic comparison with a stabilized semi-explicit method.

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The datasets generated during and/or analyszed during the current study are available from the corresponding author on reasonable request.

References

  1. Akrivis, G., Li, B., Li, D.: Energy-decaying extrapolated RK-SAV methods for the Allen–Cahn and Cahn–Hilliard equations. SIAM J. Sci. Comput. 41(6), A3703–A3727 (2019)

    MathSciNet  MATH  Google Scholar 

  2. Bandekar, A., Zhu, C., Gomez, A., Menzenski, M.Z., Sempkowski, M., Sofou, S.: Masking and triggered unmasking of targeting ligands on liposomal chemotherapy selectively suppress tumor growth in vivo. Mol. Pharm. 10(1), 152–160 (2013)

    Google Scholar 

  3. Burman, E., Hansbo, P., Larson, M.G.: A stabilized cut finite element method for partial differential equations on surfaces: the Laplace–Beltrami operator. Comput. Methods Appl. Mech. Eng. 285, 188–207 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Cahn, J.W.: On spinodal decomposition. Acta Metall. 9(9), 795–801 (1961)

    Google Scholar 

  5. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    MATH  Google Scholar 

  6. Chen, C., Yang, X.: Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn–Hilliard model. Comput. Methods Appl. Mech. Eng. 351, 35–59 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Chen, H., Mao, J., Shen, J.: Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows. Numer. Math. 145(1), 167–196 (2020)

    MathSciNet  MATH  Google Scholar 

  8. Cheng, Q., Shen, J., Yang, X.: Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach. J. Sci. Comput. 78, 1467–1487 (2019)

    MathSciNet  MATH  Google Scholar 

  9. DROPS package. http://www.igpm.rwth-aachen.de/DROPS/

  10. Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99(1), 47–84 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Fraaije, J.G.E.M., Sevink, G.J.A.: Model for pattern formation in polymer surfactant nanodroplets. Macromolecules 36(21), 7891–7893 (2003)

    Google Scholar 

  13. Garcke, H., Kampmann, J., Rätz, A., Röger, M.: A coupled surface-Cahn–Hilliard bulk–diffusion system modeling lipid raft formation in cell membranes. Math. Models Methods Appl. Sci. 26(06), 1149–1189 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Giacomelli, L., Otto, F.: Variational formulation for the lubrication approximation of the Hele–Shaw flow. Calc. Var. Partial. Differ. Equ. 13, 377–403 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Gómez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.: Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197(49–50), 4333–4352 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Grande, J., Lehrenfeld, C., Reusken, A.: Analysis of a high-order trace finite element method for PDEs on level set surfaces. SIAM J. Numer. Anal. 56(1), 228–255 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Grigorieff, R.D.: Stability of multistep-methods on variable grids. Numer. Math. 42, 359–377 (1983)

    MathSciNet  MATH  Google Scholar 

  18. Guillen-Gonzalez, F., Tierra, G.: Energy-stable and boundedness preserving numerical schemes for the Cahn–Hilliard equation with degenerate mobility. arXiv preprint arXiv: 2301.04913 (2023)

  19. Haußer, F., Li, S., Lowengrub, J., Marth, W., Rätz, A., Voigt, A.: Thermodynamically consistent models for two-component vesicles. Int. J. Biomath. Biostat 2(1), 19–48 (2013)

    MATH  Google Scholar 

  20. Huang, F., Shen, J., Yang, Z.: A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. SIAM J. Sci. Comput. 42(4), A2514–A2536 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Huang, Q.-A., Jiang, W., Yang, J.Z., Yuan, C.: Upwind-SAV approach for constructing bound-preserving and energy-stable schemes of the Cahn–Hilliard equation with degenerate mobility. arXiv preprint arXiv:2210.16017 (2022)

  22. Karve, S., Bandekar, A., Ali, M.R., Sofou, S.: The pH-dependent association with cancer cells of tunable functionalized lipid vesicles with encapsulated doxorubicin for high cell-kill selectivity. Biomaterials 31(15), 4409–4416 (2010)

    Google Scholar 

  23. Lehrenfeld, C., Olshanskii, M.A., Xu, X.: A stabilized trace finite element method for partial differential equations on evolving surfaces. SIAM J. Numer. Anal. 56(3), 1643–1672 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Li, X., Shen, J., Rui, H.: Stability and error analysis of a second-order SAV scheme with block-centered finite differences for gradient flows. Math. Comput. 88, 2047–2068 (2019)

    MATH  Google Scholar 

  25. Liao, H.-L., Ji, B., Wang, L., Zhang, Z.: Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn–Hilliard model. J. Sci. Comput. 92(2), 52 (2022)

    MathSciNet  MATH  Google Scholar 

  26. Liao, H.-L., Ji, B., Zhang, L.: An adaptive BDF2 implicit time-stepping method for the phase field crystal model. IMA J. Numer. Anal. 42(1), 649–679 (2022)

    MathSciNet  MATH  Google Scholar 

  27. Liao, H.-L., Song, X., Tang, T., Zhou, T.: Analysis of the second-order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection. Sci. China Math. 64, 887–902 (2021)

    MathSciNet  MATH  Google Scholar 

  28. Liao, H.-L., Zhang, Z.: Analysis of adaptive BDF2 scheme for diffusion equations. Math. Comput. 90(329), 1207–1226 (2021)

    MathSciNet  MATH  Google Scholar 

  29. Lisini, S., Matthes, D., Savaré, G.: Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics. J. Differ. Equ. 253(2), 814–850 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Olshanskii, M.A., Reusken, A., Grande, J.: A finite element method for elliptic equations on surfaces. SIAM J. Numer. Anal. 47, 3339–3358 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Olshanskii, M.A., Reusken, A., Xu, X.: An Eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM J. Numer. Anal. 52(3), 1354–1377 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Olshanskii, M.A., Xu, X.: A trace finite element method for PDEs on evolving surfaces. SIAM J. Sci. Comput. 39(4), A1301–A1319 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Palzhanov, Y., Zhiliakov, A., Quaini, A., Olshanskii, M.: A decoupled, stable, and linear fem for a phase-field model of variable density two-phase incompressible surface flow. Comput. Methods Appl. Mech. Eng. 387, 114167 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Qian, T., Sheng, P.: Generalized hydrodynamic equations for nematic liquid crystals. Phys. Rev. E 58, 7475–7485 (1998)

    Google Scholar 

  35. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895–2912 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. A 28, 1669 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Sun, M., Feng, X., Wang, K.: Numerical simulation of binary fluid-surfactant phase field model coupled with geometric curvature on the curved surface. Comput. Methods Appl. Mech. Eng. 367, 113123 (2020)

    MathSciNet  MATH  Google Scholar 

  40. Sun, M., Xiao, X., Feng, X., Wang, K.: Modeling and numerical simulation of surfactant systems with incompressible fluid flows on surfaces. Comput. Methods Appl. Mech. Eng. 390, 114450 (2022)

    MathSciNet  MATH  Google Scholar 

  41. Wang, M., Huang, Q., Wang, C.: A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation. J. Sci. Comput. 88(2), 33 (2021)

    MathSciNet  MATH  Google Scholar 

  42. Wang, Y., Palzhanov, Y., Quaini, A., Olshanskii, M., Majd, S.: Lipid domain coarsening and fluidity in multicomponent lipid vesicles: a continuum based model and its experimental validation. Biochimica et Biophysica Acta (BBA) - Biomembranes 1864(7), 183898 (2022)

    Google Scholar 

  43. Yushutin, V., Quaini, A., Majd, S., Olshanskii, M.: A computational study of lateral phase separation in biological membranes. Int. J. Numer. Methods Biomed. Eng. 35(3), e3181 (2019)

    MathSciNet  Google Scholar 

  44. Yushutin, V., Quaini, A., Olshanskii, M.: Numerical modeling of phase separation on dynamic surfaces. J. Comput. Phys. 407, 109126 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Zhiliakov, A., Wang, Y., Quaini, A., Olshanskii, M., Majd, S.: Experimental validation of a phase-field model to predict coarsening dynamics of lipid domains in multicomponent membranes. Biochimica et Biophysica Acta (BBA)-Biomembranes 1863(1), 183446 (2021)

    Google Scholar 

  46. Zhuang, Q., Shen, J.: Efficient SAV approach for imaginary time gradient flows with applications to one-and multi-component Bose-Einstein condensates. J. Comput. Phys. 396, 72–88 (2019)

    MathSciNet  MATH  Google Scholar 

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Funding

This work was partially supported by US National Science Foundation (NSF) through grant DMS-1953535 (PI A. Q., co-PI M. O.) and DMS-2309197 (PI M. O).

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, 3551 Cullen Blvd, Houston, Texas, 77204, USA

    Maxim Olshanskii, Yerbol Palzhanov & Annalisa Quaini

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  1. Maxim Olshanskii
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  2. Yerbol Palzhanov
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  3. Annalisa Quaini
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Correspondence to Annalisa Quaini.

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Olshanskii, M., Palzhanov, Y. & Quaini, A. A Scalar Auxiliary Variable Unfitted FEM for the Surface Cahn–Hilliard Equation. J Sci Comput 97, 57 (2023). https://doi.org/10.1007/s10915-023-02370-8

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  • DOI: https://doi.org/10.1007/s10915-023-02370-8

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