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

- 757 Downloads
- 14 Citations

## 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.Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff O.D.E’.S. SIAM J. Numer. Anal.
**14**, 1006–1021 (1977)CrossRefzbMATHMathSciNetGoogle Scholar - 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.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.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.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.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.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.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.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.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.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.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.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.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.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.Eyre, D.J.: Systems of Cahn–Hilliard equations. SIAM J. Appl. Math.
**53**, 1686–1712 (1993)CrossRefzbMATHMathSciNetGoogle Scholar - 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.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.Furihata, D.: A stable and conservative finite difference scheme for the Cahn–Hilliard equation. Numer. Math.
**87**, 675–699 (2001)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Guo, R., Xu, Y., Blanca Ayuso de Dios.: Multigrid methods of discontinuous Galerkin discretization for linear time-dependent fourth-order equations, PreprintGoogle Scholar
- 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.Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn–Hilliard fluids. J. Comput. Phys.
**193**, 511–543 (2004)CrossRefzbMATHMathSciNetGoogle Scholar - 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.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.Reed, W., Hill, T.: Triangular mesh methods for the neutrontransport equation, La-ur-73-479, Los Alamos Scientific Laboratory (1973)Google Scholar
- 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.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.Trottenberg, U., Oosterlee, C., Schller, A.: Multigrid. Academic Press, New York (2005)Google Scholar
- 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.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.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.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.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.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.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