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 & 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, (2011).   Google Scholar

[2]

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

[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.  doi: 10.1137/S0036142901384162.  Google Scholar

[4]

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

[5]

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

[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.  doi: 10.1016/j.jcp.2013.04.024.  Google Scholar

[7]

J. H. Bramble, "Multigrid Methods,", Research Notes in Mathematics Series. Chapman and Hall/CRC, (1993).   Google Scholar

[8]

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

[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.  doi: 10.1137/100784497.  Google Scholar

[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.  doi: 10.1023/A:1025722804873.  Google Scholar

[11]

P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation,", Cambridge University Press, (1989).   Google Scholar

[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.  doi: 10.4208/cicp.171211.130412a.  Google Scholar

[13]

J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods,, In, (1976), 207.   Google Scholar

[14]

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

[15]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, In J.F. Rodrigues, (1988), 1.   Google Scholar

[16]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation,, In J. W. Bullard, (1998), 1686.   Google Scholar

[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.  doi: 10.1090/S0025-5718-07-01985-0.  Google Scholar

[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.  doi: 10.1137/110827119.  Google Scholar

[19]

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

[20]

W. Hackbusch, "Multi-Grid Methods and Applications,", Springer Series in Computational Mathematics. Springer, (2010).   Google Scholar

[21]

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

[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.  doi: 10.1016/j.jcp.2009.04.020.  Google Scholar

[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.  doi: 10.1137/S0036142996297199.  Google Scholar

[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.  doi: 10.1137/080726768.  Google Scholar

[25]

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

[26]

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

[27]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid,", Academic Press, (2005).   Google Scholar

[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.  doi: 10.3934/dcds.2010.28.405.  Google Scholar

[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.  doi: 10.1137/090752675.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.03.010.  Google Scholar

[31]

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

[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.  doi: 10.1007/s10915-010-9363-4.  Google Scholar

[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.  doi: 10.1137/080738143.  Google Scholar

[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.  doi: 10.1016/j.jcp.2007.08.001.  Google Scholar

show all references

References:
[1]

A. Aristotelous, "Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth,", PhD thesis, (2011).   Google Scholar

[2]

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

[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.  doi: 10.1137/S0036142901384162.  Google Scholar

[4]

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

[5]

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

[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.  doi: 10.1016/j.jcp.2013.04.024.  Google Scholar

[7]

J. H. Bramble, "Multigrid Methods,", Research Notes in Mathematics Series. Chapman and Hall/CRC, (1993).   Google Scholar

[8]

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

[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.  doi: 10.1137/100784497.  Google Scholar

[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.  doi: 10.1023/A:1025722804873.  Google Scholar

[11]

P. G. Ciarlet, "Introduction to Numerical Linear Algebra and Optimisation,", Cambridge University Press, (1989).   Google Scholar

[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.  doi: 10.4208/cicp.171211.130412a.  Google Scholar

[13]

J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods,, In, (1976), 207.   Google Scholar

[14]

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

[15]

C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, In J.F. Rodrigues, (1988), 1.   Google Scholar

[16]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation,, In J. W. Bullard, (1998), 1686.   Google Scholar

[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.  doi: 10.1090/S0025-5718-07-01985-0.  Google Scholar

[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.  doi: 10.1137/110827119.  Google Scholar

[19]

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

[20]

W. Hackbusch, "Multi-Grid Methods and Applications,", Springer Series in Computational Mathematics. Springer, (2010).   Google Scholar

[21]

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

[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.  doi: 10.1016/j.jcp.2009.04.020.  Google Scholar

[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.  doi: 10.1137/S0036142996297199.  Google Scholar

[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.  doi: 10.1137/080726768.  Google Scholar

[25]

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

[26]

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

[27]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid,", Academic Press, (2005).   Google Scholar

[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.  doi: 10.3934/dcds.2010.28.405.  Google Scholar

[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.  doi: 10.1137/090752675.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.03.010.  Google Scholar

[31]

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

[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.  doi: 10.1007/s10915-010-9363-4.  Google Scholar

[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.  doi: 10.1137/080738143.  Google Scholar

[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.  doi: 10.1016/j.jcp.2007.08.001.  Google Scholar

[1]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[3]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[4]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[5]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[6]

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
[7]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[8]

Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049
[9]

Lunji Song, Zhimin Zhang. Polynomial preserving recovery of an over-penalized symmetric interior penalty Galerkin method for elliptic problems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1405-1426. doi: 10.3934/dcdsb.2015.20.1405
[10]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[11]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[12]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[13]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[14]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[15]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[16]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[17]

Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113
[18]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[19]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
[20]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences