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February  2021, 14(2): 653-676. doi: 10.3934/dcdss.2020353

A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions

1. 

Faculté des Sciences et Techniques, Université Marien Ngouabi, Brazzaville, Congo Brazzaville

2. 

Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86073 Poitiers, France

* Corresponding author: Morgan Pierre

Dedicated to Michel Pierre on the occasion of his 70th birthday

Received  October 2019 Revised  January 2020 Published  February 2021 Early access  May 2020

We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution is shown to converge to the energy solution of the problem as the time step tends to $ 0 $. The proof involves a multivalued operator and a monotonicity argument. This approach allows us to compute numerically singular solutions to the problem.

Citation: Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353
References:
[1]

P. F. AntoniettiM. GrasselliS. Stangalino and M. Verani, Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions, J. Sci. Comput., 66 (2016), 1260-1280.  doi: 10.1007/s10915-015-0063-y.

[2]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[4]

S. Bartels and R. Müller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential, Numer. Math., 119 (2011), 409-435.  doi: 10.1007/s00211-011-0389-9.

[5]

N. Batangouna and M. Pierre, Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system, Commun. Pure Appl. Anal., 17 (2018), 1-19.  doi: 10.3934/cpaa.2018001.

[6]

F. Boyer and F. Nabet, A DDFV method for a Cahn-Hilliard/Stokes phase field model with dynamic boundary conditions, ESAIM Math. Model. Numer. Anal., 51 (2017), 1691-1731.  doi: 10.1051/m2an/2016073.

[7]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[8]

H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827.

[10]

W. Chen, C. Wang, X. Wang and S. M. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys. X, 3 (2019). doi: 10.1016/j.jcpx.2019.100031.

[11]

L. CherfilsH. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.

[12]

L. CherfilsH. Fakih and A. Miranville, On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.

[13]

L. CherfilsS. Gatti and A. Miranville, Corrigendum to: "Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials", J. Math. Anal. Appl., 348 (2008), 1029-1030.  doi: 10.1016/j.jmaa.2008.07.058.

[14]

L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 25–34.

[15]

L. CherfilsS. Gatti and A. Miranville, Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.  doi: 10.3934/cpaa.2012.11.2261.

[16]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129. 

[17]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[18]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.

[19]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-549.  doi: 10.1007/s00211-014-0618-0.

[20]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.

[21]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.

[22]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[23]

H. P. FischerP. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896.  doi: 10.1103/PhysRevLett.79.893.

[24]

H. P. FischerP. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. 

[25]

H. P. FischerJ. ReinhardW. DieterichJ.-F. GouyetP. MaassA. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037.  doi: 10.1063/1.475690.

[26]

T. FukaoS. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case, Commun. Pure Appl. Anal., 16 (2017), 1915-1938.  doi: 10.3934/cpaa.2017093.

[27]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[28]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.  doi: 10.4171/ZAA/1277.

[29]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[30]

H. IsraelA. Miranville and M. Petcu, Numerical analysis of a Cahn-Hilliard type equation with dynamic boundary conditions, Ric. Mat., 64 (2015), 25-50.  doi: 10.1007/s11587-014-0187-7.

[31]

B. Kovács and C. Lubich, Numerical analysis of parabolic problems with dynamic boundary conditions, IMA J. Numer. Anal., 37 (2017), 1-39.  doi: 10.1093/imanum/drw015.

[32]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[33]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[34]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.

[35]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differential Equations, 2006 (2006), 1-20. 

[36]

F. Nabet, Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions, IMA J. Numer. Anal., 36 (2016), 1898-1942.  doi: 10.1093/imanum/drv057.

[37]

M. Pierre and M. Pierre, Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation, Discrete Contin. Dyn. Syst., 33 (2013), 5347-5377.  doi: 10.3934/dcds.2013.33.5347.

[38]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

[39]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984. doi: 10.1090/chel/343.

show all references

References:
[1]

P. F. AntoniettiM. GrasselliS. Stangalino and M. Verani, Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions, J. Sci. Comput., 66 (2016), 1260-1280.  doi: 10.1007/s10915-015-0063-y.

[2]

H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[4]

S. Bartels and R. Müller, Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential, Numer. Math., 119 (2011), 409-435.  doi: 10.1007/s00211-011-0389-9.

[5]

N. Batangouna and M. Pierre, Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system, Commun. Pure Appl. Anal., 17 (2018), 1-19.  doi: 10.3934/cpaa.2018001.

[6]

F. Boyer and F. Nabet, A DDFV method for a Cahn-Hilliard/Stokes phase field model with dynamic boundary conditions, ESAIM Math. Model. Numer. Anal., 51 (2017), 1691-1731.  doi: 10.1051/m2an/2016073.

[7]

H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[8]

H. Brezis, Analyse Fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.

[9]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245.  doi: 10.1007/BF00254827.

[10]

W. Chen, C. Wang, X. Wang and S. M. Wise, Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential, J. Comput. Phys. X, 3 (2019). doi: 10.1016/j.jcpx.2019.100031.

[11]

L. CherfilsH. Fakih and A. Miranville, Finite-dimensional attractors for the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation in image inpainting, Inverse Probl. Imaging, 9 (2015), 105-125.  doi: 10.3934/ipi.2015.9.105.

[12]

L. CherfilsH. Fakih and A. Miranville, On the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with logarithmic nonlinear terms, SIAM J. Imaging Sci., 8 (2015), 1123-1140.  doi: 10.1137/140985627.

[13]

L. CherfilsS. Gatti and A. Miranville, Corrigendum to: "Existence of global solutions to the Caginalp phase-field system with dynamic boundary conditions and singular potentials", J. Math. Anal. Appl., 348 (2008), 1029-1030.  doi: 10.1016/j.jmaa.2008.07.058.

[14]

L. Cherfils, S. Gatti and A. Miranville, Finite dimensional attractors for the Caginalp system with singular potentials and dynamic boundary conditions, Bull. Transilv. Univ. Braşov Ser. III, 2 (2009), 25–34.

[15]

L. CherfilsS. Gatti and A. Miranville, Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290.  doi: 10.3934/cpaa.2012.11.2261.

[16]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129. 

[17]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115.  doi: 10.1007/s10492-009-0008-6.

[18]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.  doi: 10.1007/s00032-011-0165-4.

[19]

L. Cherfils and M. Petcu, A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 517-549.  doi: 10.1007/s00211-014-0618-0.

[20]

L. CherfilsM. Petcu and M. Pierre, A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511-1533.  doi: 10.3934/dcds.2010.27.1511.

[21]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65.  doi: 10.1007/BF01385847.

[22]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.

[23]

H. P. FischerP. Maass and W. Dieterich, Novel surface modes in spinodal decomposition, Phys. Rev. Lett., 79 (1997), 893-896.  doi: 10.1103/PhysRevLett.79.893.

[24]

H. P. FischerP. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Lett., 42 (1998), 49-54. 

[25]

H. P. FischerJ. ReinhardW. DieterichJ.-F. GouyetP. MaassA. Majhofer and D. Reinel, Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall, J. Chem. Phys., 108 (1998), 3028-3037.  doi: 10.1063/1.475690.

[26]

T. FukaoS. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case, Commun. Pure Appl. Anal., 16 (2017), 1915-1938.  doi: 10.3934/cpaa.2017093.

[27]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Syst., 28 (2010), 67-98.  doi: 10.3934/dcds.2010.28.67.

[28]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72.  doi: 10.4171/ZAA/1277.

[29]

F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[30]

H. IsraelA. Miranville and M. Petcu, Numerical analysis of a Cahn-Hilliard type equation with dynamic boundary conditions, Ric. Mat., 64 (2015), 25-50.  doi: 10.1007/s11587-014-0187-7.

[31]

B. Kovács and C. Lubich, Numerical analysis of parabolic problems with dynamic boundary conditions, IMA J. Numer. Anal., 37 (2017), 1-39.  doi: 10.1093/imanum/drw015.

[32]

J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[33]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[34]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Contin. Dyn. Syst., 28 (2010), 275-310.  doi: 10.3934/dcds.2010.28.275.

[35]

D. Mugnolo and S. Romanelli, Dirichlet forms for general Wentzell boundary conditions, analytic semigroups, and cosine operator functions, Electron. J. Differential Equations, 2006 (2006), 1-20. 

[36]

F. Nabet, Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions, IMA J. Numer. Anal., 36 (2016), 1898-1942.  doi: 10.1093/imanum/drv057.

[37]

M. Pierre and M. Pierre, Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation, Discrete Contin. Dyn. Syst., 33 (2013), 5347-5377.  doi: 10.3934/dcds.2013.33.5347.

[38]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

[39]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984. doi: 10.1090/chel/343.

Figure 1.  Solution $ y_K $ for different values of $ K $
Figure 2.  Solutions of the regularized problem without constraint ($ y_K^ \varepsilon $, left) and with constraint ($ \tilde{y}_K^ \varepsilon $, right)
Figure 3.  Initial condition $ u_0 $
Figure 4.  Solution $ u(t) $ at time $ t = 0.10 $
Figure 5.  Solution $ u(t) $ at singular time $ t = 0.71 $
Figure 6.  Stationary solution ($ u(t) $ at time $ t = 5.00 $)
Figure 7.  Solution $ y\mapsto u(t, x = 2, y) $ from $ t = 0 $ to $ t = 5.00 $
Table 1.  $ L^2 $-error and ratio of consecutive errors vs time step
$ m $ (cf. time step) 0 1 2 3 4 5
$ L^2 $-error 0.0240 0.0124 0.0063 0.0032 0.0016 0.0008
ratio 1.94 1.97 1.97 2 2
$ m $ (cf. time step) 0 1 2 3 4 5
$ L^2 $-error 0.0240 0.0124 0.0063 0.0032 0.0016 0.0008
ratio 1.94 1.97 1.97 2 2
Table 2.  Normalized CPU time vs time step for the linearly implicit (LI) scheme and the doubly splitting (DS) scheme
$ m $ (cf. time step) 0 1 2 3 4 5
LI scheme 1 2 4 8 16 32
DS scheme 165 262 305 381 511 489
$ m $ (cf. time step) 0 1 2 3 4 5
LI scheme 1 2 4 8 16 32
DS scheme 165 262 305 381 511 489
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