-
Previous Article
Epidemic models with age of infection, indirect transmission and incomplete treatment
- DCDS-B Home
- This Issue
- Next Article
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 |
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
Other articles
by authors
[Back to Top]