Advertisement

C\(^0\)-Interior Penalty Discontinuous Galerkin Approximation of a Sixth-Order Cahn-Hilliard Equation Modeling Microemulsification Processes

  • Ronald H. W. HoppeEmail author
  • Christopher Linsenmann
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 47)

Abstract

Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C\(^0\) Interior Penalty Discontinuous Galerkin (C\(^0\)IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C\(^0\) conforming Lagrangian finite elements of polynomial degree \(p \ge 2\). For the semidiscretized problem we derive quasi-optimal a priori error estimates for the global discretization error in a mesh-dependent C\(^0\)IPDG norm. The semidiscretized problem represents an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order \( q \ge 2\). Numerical results show the formation of microemulsions in an oil/water system and confirm the theoretically derived convergence rates.

Notes

Acknowledgements

Ronald H. W. Hoppe acknowledges support by the NSF grants DMS-1115658, DMS-1216857, DMS-1520886 and by the German National Science Foundation DFG within the Priority Program SPP 1506.

References

  1. 1.
    Alexander R (1977) Diagonally implicit Runge-Kutta methods for stiff o.d.e’.s. SIAM J Numer Anal 14(6):1006–1021MathSciNetCrossRefGoogle Scholar
  2. 2.
    Boyarkin O, Hoppe RHW, Linsenmann C (2015) High order approximations in space and time of a sixth order Cahn-Hilliard equation. Russ J Numer Anal Math Model 30(6):313–328MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brenner SC, Gudi T, Sung L-Y (2010) An a posteriori error estimator for a quadratic \(C^0\)-interior penalty method for the biharmonic problem. IMA J Numer Anal 30(3):777–798MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brenner SC, Scott LR (2008) The mathematical theory of finite element methods, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  5. 5.
    Brenner SC, Sung L-Y (2005) \(c^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J Sci Comput 22(23):83–118MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brenner SC, Wang K, Zhao J (2004) Poincaré-Friedrichs inequalities for piecewise \(H^2\) functions. Numer Funct Anal Optim 25(5–6):463–478MathSciNetCrossRefGoogle Scholar
  7. 7.
    Butcher JC (2008) Numerical methods for ordinary differential equations, 2nd edn. Wiley, ChichesterCrossRefGoogle Scholar
  8. 8.
    Ciarlet PG (2002) The finite element method for elliptic problems. SIAM, Philadelphia, PACrossRefGoogle Scholar
  9. 9.
    Deuflhard P (2004) Newton methods for nonlinear problems: affine invariance and adaptive algorithms. Springer, BerlinzbMATHGoogle Scholar
  10. 10.
    Engel G, Garikipati K, Hughes TJR, Larson MG, Mazzei L, Taylor RL (2002) Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Eng 191(34):3669–3750MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fraunholz T, Hoppe RHW, Peter M (2015) Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem. J Numer Math 23(4):317–330MathSciNetCrossRefGoogle Scholar
  12. 12.
    Georgoulis EH, Houston P (2009) Discontinuous Galerkin methods for the biharmonic problem. IMA J Numer Anal 29(3):573–594MathSciNetCrossRefGoogle Scholar
  13. 13.
    Georgoulis EH, Houston P, Virtanen J (2011) An a posteriori error indicator for discontinuous Galerkin approximations of fourth order elliptic problems. IMA J Numer Anal 31(1):281–298MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ghosh PK, Murthy RS (2006) Microemulsions: a potential drug delivery system. Curr Drug Deliv 3(2):167–180CrossRefGoogle Scholar
  15. 15.
    Gompper G, Goos J (1994) Fluctuating interfaces in microemulsion and sponge phases. Phys. Rev. E 50(2):1325–1335CrossRefGoogle Scholar
  16. 16.
    Gompper G, Kraus M (1993) Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations. Phys Rev E 47(6):4289–4300CrossRefGoogle Scholar
  17. 17.
    Gompper G, Kraus M (1993) Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations. Phys Rev E 47(6):4301–4312CrossRefGoogle Scholar
  18. 18.
    Gompper G, Zschocke S (1992) Ginzburg-Landau theory of oil-water-surfactant mixtures. Phys Rev A 46(8):4836–4851CrossRefGoogle Scholar
  19. 19.
    Hairer E, Wanner G (1996) Solving ordinary differential equations. II: stiff and differential-algebraic problems, 2nd edn. Springer, BerlinGoogle Scholar
  20. 20.
    Hoppe RHW, Linsenmann C (2012) An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method. J Comput Phys 231(14):4676–4693MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jha SK, Karki R, Venkatesh DP, Geethalakshami A (2011) Formulation development and characterization of microemulsion drug delivery systems containing antiulcer drug. Int J Drug Dev Res 3(4):336–343Google Scholar
  22. 22.
    Mehta SK, Kaur G (2011) Microemulsions: thermodynamics and dynamic properties. In: Tadashi M (ed) Thermodynamics. InTech, pp 381–406. http://www.intechopen.com/books/thermodynamics/microemulsions-thermodynamic-and-dynamic-properties
  23. 23.
    Moulik SP, Rakshit AK (2006) Physiochemistry and applications of micro-emulsions. J Surf Sci Tech 22(3–4):159–186Google Scholar
  24. 24.
    Pawlow I, Zajaczkowski WM (2011) A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Commun Pure Appl Anal 10(6):1823–1847MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pawlow I, Zajaczkowski WM (2013) On a class of sixth order viscous Cahn-Hilliard type equations. Discrete Contin Dyn Syst Ser S 6(2):517–546MathSciNetCrossRefGoogle Scholar
  26. 26.
    Prince LM (1977) Microemulsions: theory and practice. Academic Press, New YorkGoogle Scholar
  27. 27.
    Rosano HL, Clausse M (eds) (1987) Microemulsion systems. Marcel Dekker, New YorkGoogle Scholar
  28. 28.
    Schimperna G, Pawlow I (2013) On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J Math Anal 45(1):31–63MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tartar L (2007) Introduction to Sobolev spaces and interpolation spaces. UMI, Bologna, Springer, BerlinzbMATHGoogle Scholar
  30. 30.
    Warburton T, Hesthaven JS (2003) On the constants in \(hp\)-finite element trace inverse inequalities. Comput Methods Appl Mech Eng 192(25):2765–2773MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wells GN, Kuhl E, Garikipati K (2006) A discontinuous Galerkin method for the Cahn-Hilliard equation. J Comput Phys 218(2):860–877MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Institute of MathematicsUniversity of AugsburgAugsburgGermany

Personalised recommendations