American Institute of Mathematical Sciences

Advanced
November  2013, 18(9): 2211-2238. doi: 10.3934/dcdsb.2013.18.2211

A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver

Andreas C. Aristotelous 1, , Ohannes Karakashian 2, and  Steven M. Wise 3,

1. 

Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC 27709, United States
2. 

Department of Mathematics, University of Tennessee, Knoxville, TN 37996, United States
3. 

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-0612, United States

Received  June 2013 Revised  August 2013 Published  September 2013

In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.
Keywords: convergence, discontinuous Galerkin, diblock copolymer, symmetric interior penalty, nonlinear multigrid., convex splitting, Cahn-Hilliard equation, energy stability.
Mathematics Subject Classification: Primary: 65M60, 65M12; Secondary: 65M5.
Citation: Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211
References:
[1]

A. Aristotelous, "Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth," PhD thesis, University of Tennessee, 2011.

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 19 (1982), 742-760. doi: 10.1137/0719052.

[3]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.

[4]

I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty, SIAM Journal on Numerical Analysis, 10 (1973), 863-875. doi: 10.1137/0710071.

[5]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31:45-59, 1977. doi: 10.1090/S0025-5718-1977-0431742-5.

[6]

A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S. M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, Journal of Computational Physics, 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024.

[7]

J. H. Bramble, "Multigrid Methods," Research Notes in Mathematics Series. Chapman and Hall/CRC, London, 1993.

[8]

J. W. Cahn, On spinodal decomposition, Acta Metallurgica, 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[9]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362. doi: 10.1137/100784497.

[10]

R. Choksi and X. Ren, On a derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176. doi: 10.1023/A:1025722804873.

[11]

P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation," Cambridge University Press, Cambridge, UK, 1989.

[12]

C. Collins, J. Shen and S. M. Wise, Unconditionally stable finite difference multigrid schemes for the Cahn-Hilliard-Brinkman equation, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a.

[13]

J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, In "Computing Methods in Applied Sciences," pages 207-216. Springer, Berlin, 1976.

[14]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Archive for Rational Mechanics and Analysis, 96 (1986) ,339-357. doi: 10.1007/BF00251803.

[15]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, In J.F. Rodrigues, editor, Mathematical Models for Phase Change Problems: Proceedings of the European Workshop held at Óbidos, Portugal, October 1-3, 1988, International Series of Numerical Mathematics, 35-73, Berlin, 1989. Birkhäuser Verlag.

[16]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In J. W. Bullard, R. Kalia, M. Stoneham, and L.Q. Chen, editors, Computational and Mathematical Models of Microstructural Evolution, volume 53, pages 1686-1712, Warrendale, PA, USA, 1998. Materials Research Society.

[17]

X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comput., 76 (2007), 1093-1117. doi: 10.1090/S0025-5718-07-01985-0.

[18]

X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM Journal on Numerical Analysis, 50 (2012), 1320-1343. doi: 10.1137/110827119.

[19]

J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numerische Mathematik, 95 (2003), 527-550. doi: 10.1007/s002110200392.

[20]

W. Hackbusch, "Multi-Grid Methods and Applications," Springer Series in Computational Mathematics. Springer, Berlin, 2010.

[21]

M. R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM Journal on Numerical Analysis, 30 (1993), 184-214. doi: 10.1137/0730009.

[22]

Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, Journal of Computational Physics, 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020.

[23]

O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 35 (1998), 93-120. doi: 10.1137/S0036142996297199.

[24]

D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM Journal on Numerical Analysis, 47 (2009), 2660-2685. doi: 10.1137/080726768.

[25]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[26]

P. Percell and M. F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM Journal on Numerical Analysis, 15 (1978), 705-714. doi: 10.1137/0715047.

[27]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, New York, 2005.

[28]

C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[29]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM Journal on Numerical Analysis, 49 (2011), 945-969. doi: 10.1137/090752675.

[30]

G. N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation, Journal of Computational Physics, 218 (2006), 860-877. doi: 10.1016/j.jcp.2006.03.010.

[31]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010.

[32]

S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, Journal of Scientific Computing, 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.

[33]

S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis, 47 (2009), 2269-2288. doi: 10.1137/080738143.

[34]

Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, Journal of Computational Physics, 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.

show all references

References:
[1]

A. Aristotelous, "Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth," PhD thesis, University of Tennessee, 2011.

[2]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 19 (1982), 742-760. doi: 10.1137/0719052.

[3]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), 1749-1779. doi: 10.1137/S0036142901384162.

[4]

I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty, SIAM Journal on Numerical Analysis, 10 (1973), 863-875. doi: 10.1137/0710071.

[5]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31:45-59, 1977. doi: 10.1090/S0025-5718-1977-0431742-5.

[6]

A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S. M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation, Journal of Computational Physics, 250 (2013), 270-292. doi: 10.1016/j.jcp.2013.04.024.

[7]

J. H. Bramble, "Multigrid Methods," Research Notes in Mathematics Series. Chapman and Hall/CRC, London, 1993.

[8]

J. W. Cahn, On spinodal decomposition, Acta Metallurgica, 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.

[9]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362. doi: 10.1137/100784497.

[10]

R. Choksi and X. Ren, On a derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176. doi: 10.1023/A:1025722804873.

[11]

P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation," Cambridge University Press, Cambridge, UK, 1989.

[12]

C. Collins, J. Shen and S. M. Wise, Unconditionally stable finite difference multigrid schemes for the Cahn-Hilliard-Brinkman equation, Commun. Comput. Phys., 13 (2013), 929-957. doi: 10.4208/cicp.171211.130412a.

[13]

J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, In "Computing Methods in Applied Sciences," pages 207-216. Springer, Berlin, 1976.

[14]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Archive for Rational Mechanics and Analysis, 96 (1986) ,339-357. doi: 10.1007/BF00251803.

[15]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, In J.F. Rodrigues, editor, Mathematical Models for Phase Change Problems: Proceedings of the European Workshop held at Óbidos, Portugal, October 1-3, 1988, International Series of Numerical Mathematics, 35-73, Berlin, 1989. Birkhäuser Verlag.

[16]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, In J. W. Bullard, R. Kalia, M. Stoneham, and L.Q. Chen, editors, Computational and Mathematical Models of Microstructural Evolution, volume 53, pages 1686-1712, Warrendale, PA, USA, 1998. Materials Research Society.

[17]

X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition, Math. Comput., 76 (2007), 1093-1117. doi: 10.1090/S0025-5718-07-01985-0.

[18]

X. Feng and S. M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation, SIAM Journal on Numerical Analysis, 50 (2012), 1320-1343. doi: 10.1137/110827119.

[19]

J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numerische Mathematik, 95 (2003), 527-550. doi: 10.1007/s002110200392.

[20]

W. Hackbusch, "Multi-Grid Methods and Applications," Springer Series in Computational Mathematics. Springer, Berlin, 2010.

[21]

M. R. Hanisch, Multigrid preconditioning for the biharmonic Dirichlet problem, SIAM Journal on Numerical Analysis, 30 (1993), 184-214. doi: 10.1137/0730009.

[22]

Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation, Journal of Computational Physics, 228 (2009), 5323-5339. doi: 10.1016/j.jcp.2009.04.020.

[23]

O. A. Karakashian and W. N. Jureidini, A nonconforming finite element method for the stationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 35 (1998), 93-120. doi: 10.1137/S0036142996297199.

[24]

D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection, SIAM Journal on Numerical Analysis, 47 (2009), 2660-2685. doi: 10.1137/080726768.

[25]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. doi: 10.1021/ma00164a028.

[26]

P. Percell and M. F. Wheeler, A local residual finite element procedure for elliptic equations, SIAM Journal on Numerical Analysis, 15 (1978), 705-714. doi: 10.1137/0715047.

[27]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid," Academic Press, New York, 2005.

[28]

C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[29]

C. Wang and S. M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM Journal on Numerical Analysis, 49 (2011), 945-969. doi: 10.1137/090752675.

[30]

G. N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation, Journal of Computational Physics, 218 (2006), 860-877. doi: 10.1016/j.jcp.2006.03.010.

[31]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM Journal on Numerical Analysis, 15 (1978), 152-161. doi: 10.1137/0715010.

[32]

S. M. Wise, Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn-Hilliard-Hele-Shaw system of equations, Journal of Scientific Computing, 44 (2010), 38-68. doi: 10.1007/s10915-010-9363-4.

[33]

S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM Journal on Numerical Analysis, 47 (2009), 2269-2288. doi: 10.1137/080738143.

[34]

Y. Xia, Y. Xu and C. W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations, Journal of Computational Physics, 227 (2007), 472-491. doi: 10.1016/j.jcp.2007.08.001.

[1]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[2]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[3]

Matthieu Brachet, Philippe Parnaudeau, Morgan Pierre. Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1987-2031. doi: 10.3934/dcdss.2022110
[4]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
[5]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[6]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[7]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[8]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024
[9]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[10]

Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049
[11]

Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405
[12]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
[13]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[14]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[15]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[16]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[17]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[18]

Hussein Fakih, Ragheb Mghames, Noura Nasreddine. On the Cahn-Hilliard equation with mass source for biological applications. Communications on Pure and Applied Analysis, 2021, 20 (2) : 495-510. doi: 10.3934/cpaa.2020277
[19]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[20]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (326)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy