# American Institute of Mathematical Sciences

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

 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.
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.

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##### 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.
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