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A Robust Solver for a Mixed Finite Element Method for the Cahn–Hilliard Equation

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Abstract

We develop a robust solver for a mixed finite element convex splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.

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References

  1. Alnæs, M.S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 15. Arch. Numer. Soft. 3, 9–23 (2015)

    Google Scholar 

  2. Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. 65, 301–314 (2013)

    Article  MathSciNet  Google Scholar 

  3. Barrett, J.W., Nurnberg, R., Styles, V.: Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 42(2), 738–772 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bänsch, E., Morin, P., Nochetto, R.H.: Preconditioning a class of fourth order problems by operator splitting. Numer. Math. 118, 197–228 (2011)

    Article  MathSciNet  Google Scholar 

  5. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  Google Scholar 

  6. Boyanova, P., Do-Quang, M., Neytcheva, M.: Efficient preconditioners for large scale binary Cahn–Hilliard models. Comput. Methods Appl. Math. 12(1), 1–22 (2012)

    Article  MathSciNet  Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

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

    Article  Google Scholar 

  9. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258 (1958)

    Article  Google Scholar 

  10. Ceniceros, H.D., Roma, A.M.: A nonstiff, adaptive mesh refinement-based method for the Cahn–Hilliard equation. J. Comput. Phys. 225, 1849–1862 (2007)

    Article  MathSciNet  Google Scholar 

  11. Choksi, R., Maras, M., Williams, J.F.: 2D phase diagram for minimizers of a Cahn–Hilliard functional with long-range interactions. SIAM J. Appl. Dyn. Sys. 10(4), 1344–1362 (2011)

    Article  MathSciNet  Google Scholar 

  12. Diegel, A., Feng, X., Wise, S.M.: Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system. SIAM J. Numer. Anal. 53(1), 127–152 (2015)

    Article  MathSciNet  Google Scholar 

  13. Diegel, A., Wang, C., Wise, S.M.: Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation. IMA J. Numer. Anal. 36, 1867–1897 (2016)

    Article  MathSciNet  Google Scholar 

  14. Du, Q., Li, M., Liu, C.: Analysis of a phase field Navier–Stokes vesicle–fluid interaction model. Discrete Contin. Dyn. Syst. Ser. B 8(3), 539 (2007)

    Article  MathSciNet  Google Scholar 

  15. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)

    Book  Google Scholar 

  16. Elliott, C.M., Zheng, S.: On the Cahn–Hilliard equation. Arch. Ration. Mech. Anal. 96, 339–357 (1986)

    Article  MathSciNet  Google Scholar 

  17. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014)

    Book  Google Scholar 

  18. Eyre, D.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Bullard, J.W., Kalia, R., Stoneham, M., Chen, L.Q. (eds.) Computational and Mathematical Models of Microstructural Evolution, vol. 53, pp. 1686–1712. Materials Research Society, Warrendale (1998)

    Google Scholar 

  19. Feng, X.: Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1049–1072 (2006)

    Article  MathSciNet  Google Scholar 

  20. Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)

    Book  Google Scholar 

  21. Hackbusch, W.: Multi-grid Methods and Applications. Springer, Berlin (1985)

    Book  Google Scholar 

  22. Kay, D., Welford, R.: A multigrid finite element solver for the Cahn–Hilliard equation. J. Comput. Phys. 212, 288–304 (2006)

    Article  MathSciNet  Google Scholar 

  23. Kim, J.: A numerical method for the Cahn–Hilliard equation with a variable mobility. Commun. Nonlinear Sci. Numer. Simul. 12, 1560–1571 (2007)

    Article  MathSciNet  Google Scholar 

  24. Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193, 511–543 (2004)

    Article  MathSciNet  Google Scholar 

  25. Lee, C., Jeong, D., Shin, J., Li, Y., Kim, J.: A fourth-order spatial accurate and practically stable compact scheme for the Cahn–Hilliard equation. Physics A 409, 17–28 (2014)

    Article  MathSciNet  Google Scholar 

  26. Lee, D., Huh, J., Jeong, D., Shin, J., Yun, A., Kim, J.: Physical, mathematical, and numerical derivations of the Cahn–Hilliard equation. Comput. Mater. Sci. 81, 216–225 (2014)

    Article  Google Scholar 

  27. Lee, H.G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele–Shaw cell. I. The models and their calibration. Phys. Fluids 14, 492–513 (2002)

    Article  MathSciNet  Google Scholar 

  28. Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18, 1–40 (2011)

    Article  MathSciNet  Google Scholar 

  29. Shin, J., Kim, S., Lee, D., Kim, J.: A parallel multigrid method of the Cahn–Hilliard equation. Comput. Mater. Sci. 71, 89–96 (2013)

    Article  Google Scholar 

  30. Stogner, R.H., Carey, G.F., Murray, B.T.: Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements. Int. J. Numer. Methods Eng. 76, 636–661 (2008)

    Article  Google Scholar 

  31. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1988)

    Book  Google Scholar 

  32. Tierra, G., Guillén-González, F.: Numerical methods for solving the Cahn–Hilliard equation and its applicability to related energy-based models. Arch. Comput. Methods Eng. 22, 269–289 (2015)

    Article  MathSciNet  Google Scholar 

  33. Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, San Diego (2001)

    MATH  Google Scholar 

  34. van Teeffelen, S., Backofen, R., Voigt, A., Löwen, H.: Derivation of the phase-field-crystal model for colloidal solidification. Phys. Rev. E 79, 051404 (2009)

    Article  Google Scholar 

  35. Walker, S.W.: FELICITY: finite element implementation and computational interface tool for you. http://www.mathworks.com/matlabcentral/fileexchange/31141-felicity

  36. Wang, W., Chen, L., Zhou, J.: Postprocessing mixed finite element methods for solving Cahn–Hilliard equation: methods and error analysis. J. Sci. Comput. 67, 724–746 (2016)

    Article  MathSciNet  Google Scholar 

  37. Wise, S.M.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn–Hilliard–Hele–Shaw system of equations. J. Sci. Comput. 44, 38–68 (2010)

    Article  MathSciNet  Google Scholar 

  38. Wise, S.M., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn–Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226, 414–446 (2007)

    Article  MathSciNet  Google Scholar 

  39. Zulehner, W.: Nonstandard norms and robust estimates for saddle point problems. SIAM J. Matrix Anal. Appl. 32(2), 536–560 (2011)

    Article  MathSciNet  Google Scholar 

  40. Zheng, B., Chen, L.-P., Hu, X., Chen, L., Nochetto, R.H., Xu, J.: Fast multilevel solvers for a class of discrete fourth order parabolic problems. J. Sci. Comput. 69, 201–226 (2016)

    Article  MathSciNet  Google Scholar 

  41. Zhou, J., Chen, L., Huang, Y., Wang, W.: An efficient two-grid scheme for the Cahn–Hilliard equation. Commun. Comput. Phys. 17, 127–145 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Portions of this research were conducted with high performance computational resources provided by Louisiana State University (http://www.hpc.lsu.edu). We would also like to thank Shawn Walker for his valuable advice regarding the FELICITY/C++ Toolbox for MATLAB.

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Authors and Affiliations

  1. Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, 70803, USA

    Susanne C. Brenner, Amanda E. Diegel & Li-Yeng Sung

Authors
  1. Susanne C. Brenner
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  2. Amanda E. Diegel
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  3. Li-Yeng Sung
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Corresponding author

Correspondence to Susanne C. Brenner.

Additional information

The work of the first and third authors was supported in part by the National Science Foundation under Grant No. DMS-16-20273.

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Brenner, S.C., Diegel, A.E. & Sung, LY. A Robust Solver for a Mixed Finite Element Method for the Cahn–Hilliard Equation. J Sci Comput 77, 1234–1249 (2018). https://doi.org/10.1007/s10915-018-0753-3

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  • DOI: https://doi.org/10.1007/s10915-018-0753-3

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