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Journal of Scientific Computing

, Volume 58, Issue 2, pp 380–408 | Cite as

Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations

  • Ruihan Guo
  • Yan Xu
Article

Abstract

In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn–Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods for the CH equation will require severe time step restriction \((\varDelta t \sim O(\varDelta x^4))\), so implicit methods are used to remove time step restriction. Implicit methods will result in large system of algebraic equations and a fast solver is essential. The multigrid (MG) method is used to solve the algebraic equations efficiently. The Local Mode Analysis method is used to analyze the convergence behavior of the linear MG method. The discrete energy stability for the CH equations with a special homogeneous free energy density \(\Psi (u)=\frac{1}{4}(1-u^2)^2\) is proved based on the convex splitting method. We show that the number of iterations is independent of the problem size. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. We numerically show the optimal complexity of the MG solver for \(\mathcal{P }^1\) element. For \(\mathcal{P }^2\) approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.

Keywords

Cahn–Hilliard equation Local discontinuous Galerkin method  Convex splitting method Multigrid algorithm FAS multigrid Additive Runge–Kutta  Diagonally implicit Runge–Kutta Local mode analysis 

Mathematics Subject Classification (2010)

65M60 35K55 

References

  1. 1.
    Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff O.D.E’.S. SIAM J. Numer. Anal. 14, 1006–1021 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barrett, J.W., Blowey, J.F.: Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77, 1–34 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barrett, J.W., Blowey, J.F., Garcke, H.: Finite element approximation of the Cahn–Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37, 286–318 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Barrett, J.W., Blowey, J.F., Garcke, H.: On fully practical finite element approximations of degenerate Cahn–Hilliard systems. M2ANMath. Model. Numer. Anal. 35, 713–748 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bassi, F., Ghidoni, A., Rebay, S., Tesini, P.: High-order accurate \(p\)-multigrid discontinuous Galerkin solution of the Euler equations. Int. J. Numer. Methods Fluids 60, 847–865 (2008)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bassi, F., Ghidoni, A., Rebay, S.: Optimal Runge–Kutta smoothers for the \(p\)-multigrid discontinuous Galerkin solution of the 1D Euler equations. J. Comput. Phys. 230, 4153–4175 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Blowey, J.F., Elliott, C.M.: The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. II. Numerical analysis. Eur. J. Appl. Math. 3, 147–179 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Blowey, J.F., Copetti, M.I.M., Elliott, C.M.: Numerical analysis of a model for phase separation of multi-component alloy. IMA J. Numer. Anal. 16, 111–139 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studien [GMD Studies], 85. Gesellschaft für Mathematik und Datenverarbeitung mbH, St. Augustin (1984)Google Scholar
  10. 10.
    Brandt, A.: Rigorous quantitative analysis of multigrid. I. Constant coefficients two-level cycle with \(L_2\)-norm. SIAM J. Numer. Anal. 31, 1695–730 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Choo, S.M., Lee, Y.J.: A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Appl. Math. Comput. 18, 113–126 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn–Hilliard equation with logarithmic free energy. Numer. Math. 63, 39–65 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Elliott, C.M., French, D.A.: A nonconforming finite-element method for the two-dimensional Cahn–Hilliard equation. SIAM J. Numer. Anal. 26, 884–903 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn–Hilliard equation. Numer. Math. 54, 575–590 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Eyre, D.J.: Systems of Cahn–Hilliard equations. SIAM J. Appl. Math. 53, 1686–1712 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Feng, X., Prohl, A.: Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73, 541–567 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Feng, X.B., Karakashian, O.A.: Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn–Hilliard equation of phase transition. Math. Comput. 76, 1093–1117 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Furihata, D.: A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math. 87, 675–699 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hector, Gomez B., Calo, Victor M., Hughes Thomas, J.R.: Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197, 4333–4352 (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, R., Xu, Y., Blanca Ayuso de Dios.: Multigrid methods of discontinuous Galerkin discretization for linear time-dependent fourth-order equations, PreprintGoogle Scholar
  22. 22.
    Kay, D., Welford, R.: A multigrid finite element solver for the Cahn–Hilliard equation. J. Comput. Phys. 212, 288–304 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys. 193, 511–543 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for ternary Cahn–Hilliard systems. Commun. Math. Sci. 2, 53–77 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Klaij, C.M., van Raalte, M.H., van der Ven, H., van der Vegt, J.J.W.: \(h\)-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 227, 1024–1045 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Reed, W., Hill, T.: Triangular mesh methods for the neutrontransport equation, La-ur-73-479, Los Alamos Scientific Laboratory (1973)Google Scholar
  27. 27.
    Shahbazi, K., Mavriplis, D.J., Burgess, N.K.: Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 228, 7917–7940 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Sun, Z.Z.: A second-order accurate linearized difference scheme for the twodimensional Cahn–Hilliard equation. Math. Comput. 64, 1463–1471 (1995)zbMATHGoogle Scholar
  29. 29.
    Trottenberg, U., Oosterlee, C., Schller, A.: Multigrid. Academic Press, New York (2005)Google Scholar
  30. 30.
    van der Vegt, J.J.W., Rhebergen, S.: \(hp\)-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel analysis. J. Comput. Phys. 231, 7537–7563 (2012)CrossRefMathSciNetGoogle Scholar
  31. 31.
    van der Vegt, J.J.W., Rhebergen, S.: \(hp\)-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge–Kutta smoother. J. Comput. Phys. 231, 7564–7583 (2012)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Wells, G.N., Kuhl, E., Garikipati, K.: A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Comput. Phys. 218, 860–877 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Xia, Y., Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Cahn–Hilliard type equations. J. Comput. Phys. 227, 472–491 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Xia, Y., Xu, Y., Shu, C.-W.: Efficient time discretization for local discontinuous Galerkin methods. Discret. Contin. Dyn. Syst. Ser. B 8, 677–693 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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