| 1. | Lebanese International University, School of Arts and Sciences, Department of Mathematics and Physics, Bekaa campus, Lebanon |
| 2. | Lebanese University, Faculty of Sciences, Department of Mathematics, Houch el Oumara, Zahle, Lebanon |
| 3. | Politehnica University of Bucharest, Splaiul Independentei 313, 060042, Bucharest, Romania |
This article deals with some generalizations of the Cahn–Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn–Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn–Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.
| [1] |
A. C. Aristotelous, O. A. Karakashian and S. M. Wise,
Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source, IMA J. Numer. Anal, 35 (2015), 1167-1198.
doi: 10.1093/imanum/dru035. |
| [2] |
A. Bertozzi, S. Esedoglu and A. Gillette,
Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Imaging Proc., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
| [3] |
A. Bertozzi, S. Esedoglu and A. Gillette,
Analysis of a two-scale Cahn–Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.
doi: 10.1137/060660631. |
| [4] |
M. Burger, L. He and C. Schönlieb,
Cahn–Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 3 (2009), 1129-1167.
doi: 10.1137/080728548. |
| [5] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
| [6] |
L. Cherfils, H. Fakih and A. Miranville,
Finite-dimensional attractors for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation in image inpainting, Inv. Prob. Imaging, 9 (2015), 105-125.
doi: 10.3934/ipi.2015.9.105. |
| [7] |
L. Cherfils, H. Fakih and A. Miranville,
On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms, SIAM J. Imag. Sci., 8 (2015), 1123-1140.
doi: 10.1137/140985627. |
| [8] |
L. Cherfils, H. Fakih and A. Miranville,
A Cahn–Hilliard system with a fidelity term for color image inpainting, J. Math. Imaging Vis., 54 (2016), 117-131.
doi: 10.1007/s10851-015-0593-9. |
| [9] |
L. Cherfils, H. Fakih and A. Miranville,
A complex version of the Cahn-Hilliard equation for grayscale image inpainting, J. Multiscale Model. Simul., 15 (2017), 575-605.
doi: 10.1137/15M1040177. |
| [10] |
L. Cherfils, A. 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. |
| [11] |
L. Cherfils, A. Miranville and S. Zelik,
On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn.-B, 19 (2014), 2013-2026.
doi: 10.3934/dcdsb.2014.19.2013. |
| [12] |
L. Cherfils, M. Petcu and M. Pierre,
A numerical analysis of the Cahn–Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. S., 27 (2010), 1511-1533.
doi: 10.3934/dcds.2010.27.1511. |
| [13] |
I. C. Dolcetta, S. F. Vita and R. March,
Area-preserving curve-shortening flows: from phase separation to image processing, Interface. Free Bound., 4 (2002), 325-343.
doi: 10.4171/IFB/64. |
| [14] |
C. M. Elliott, D. A. French and F. A. Milner,
A second order splitting method for the Cahn–Hilliard equation, Numer. Math., 54 (1989), 575-590.
doi: 10.1007/BF01396363. |
| [15] |
A. Ern and J. L. Guermond, Elements finis: theorie, applications, mise en oeuvre, Springer-Verlag, Berlin, 2002. |
| [16] |
H. Fakih,
Asymptotic behavior of a generalized Cahn–Hilliard, equation with a mass source, Appl. Anal., 96 (2016), 324-348.
doi: 10.1080/00036811.2015.1135241. |
| [17] |
H. Fakih,
A Cahn–Hilliard equation with a proliferation term for biological and chemical applications, Asympt. Anal., 94 (2015), 71-104.
doi: 10.3233/ASY-151306. |
| [18] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
| [19] |
E. Khain and L. M. Sander,
A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 51-129.
|
| [20] |
A. Miranville,
Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comp., 1 (2011), 523-536.
|
| [21] |
A. Miranville,
Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation, Appl. Anal., 92 (2013), 1308-1321.
doi: 10.1080/00036811.2012.671301. |
| [22] |
A. Miranville,
Existence of solutions to a Cahn–Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.
doi: 10.1007/s00009-018-1284-8. |
| [23] |
A. Miranville,
The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.
|
| [24] |
A. Novick-Cohen and L. A. Segal,
Nonlinear Cahn–Hiliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.
|
| [25] |
Y. Oono and S. Puri,
Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.
|
| [26] |
C. B. Schönlieb and A. Bertozzi,
Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), 413-457.
|
| [27] |
S. Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis, Habilitation thesis, Université Bordeaux 1, 2010. |
show all references
| [1] |
A. C. Aristotelous, O. A. Karakashian and S. M. Wise,
Adaptive, second order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source, IMA J. Numer. Anal, 35 (2015), 1167-1198.
doi: 10.1093/imanum/dru035. |
| [2] |
A. Bertozzi, S. Esedoglu and A. Gillette,
Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Imaging Proc., 16 (2007), 285-291.
doi: 10.1109/TIP.2006.887728. |
| [3] |
A. Bertozzi, S. Esedoglu and A. Gillette,
Analysis of a two-scale Cahn–Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), 913-936.
doi: 10.1137/060660631. |
| [4] |
M. Burger, L. He and C. Schönlieb,
Cahn–Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 3 (2009), 1129-1167.
doi: 10.1137/080728548. |
| [5] |
J. W. Cahn and J. E. Hilliard,
Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
|
| [6] |
L. Cherfils, H. Fakih and A. Miranville,
Finite-dimensional attractors for the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation in image inpainting, Inv. Prob. Imaging, 9 (2015), 105-125.
doi: 10.3934/ipi.2015.9.105. |
| [7] |
L. Cherfils, H. Fakih and A. Miranville,
On the Bertozzi–Esedoglu–Gillette–Cahn–Hilliard equation with logarithmic nonlinear terms, SIAM J. Imag. Sci., 8 (2015), 1123-1140.
doi: 10.1137/140985627. |
| [8] |
L. Cherfils, H. Fakih and A. Miranville,
A Cahn–Hilliard system with a fidelity term for color image inpainting, J. Math. Imaging Vis., 54 (2016), 117-131.
doi: 10.1007/s10851-015-0593-9. |
| [9] |
L. Cherfils, H. Fakih and A. Miranville,
A complex version of the Cahn-Hilliard equation for grayscale image inpainting, J. Multiscale Model. Simul., 15 (2017), 575-605.
doi: 10.1137/15M1040177. |
| [10] |
L. Cherfils, A. 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. |
| [11] |
L. Cherfils, A. Miranville and S. Zelik,
On a generalized Cahn–Hilliard equation with biological applications, Discrete Cont. Dyn.-B, 19 (2014), 2013-2026.
doi: 10.3934/dcdsb.2014.19.2013. |
| [12] |
L. Cherfils, M. Petcu and M. Pierre,
A numerical analysis of the Cahn–Hilliard equation with dynamic boundary conditions, Discrete Cont. Dyn. S., 27 (2010), 1511-1533.
doi: 10.3934/dcds.2010.27.1511. |
| [13] |
I. C. Dolcetta, S. F. Vita and R. March,
Area-preserving curve-shortening flows: from phase separation to image processing, Interface. Free Bound., 4 (2002), 325-343.
doi: 10.4171/IFB/64. |
| [14] |
C. M. Elliott, D. A. French and F. A. Milner,
A second order splitting method for the Cahn–Hilliard equation, Numer. Math., 54 (1989), 575-590.
doi: 10.1007/BF01396363. |
| [15] |
A. Ern and J. L. Guermond, Elements finis: theorie, applications, mise en oeuvre, Springer-Verlag, Berlin, 2002. |
| [16] |
H. Fakih,
Asymptotic behavior of a generalized Cahn–Hilliard, equation with a mass source, Appl. Anal., 96 (2016), 324-348.
doi: 10.1080/00036811.2015.1135241. |
| [17] |
H. Fakih,
A Cahn–Hilliard equation with a proliferation term for biological and chemical applications, Asympt. Anal., 94 (2015), 71-104.
doi: 10.3233/ASY-151306. |
| [18] |
F. Hecht,
New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
doi: 10.1515/jnum-2012-0013. |
| [19] |
E. Khain and L. M. Sander,
A generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 51-129.
|
| [20] |
A. Miranville,
Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comp., 1 (2011), 523-536.
|
| [21] |
A. Miranville,
Asymptotic behavior of a generalized Cahn–Hilliard equation with a proliferation, Appl. Anal., 92 (2013), 1308-1321.
doi: 10.1080/00036811.2012.671301. |
| [22] |
A. Miranville,
Existence of solutions to a Cahn–Hilliard type equation with a logarithmic nonlinear term, Mediterr. J. Math., 16 (2019), 1-18.
doi: 10.1007/s00009-018-1284-8. |
| [23] |
A. Miranville,
The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.
|
| [24] |
A. Novick-Cohen and L. A. Segal,
Nonlinear Cahn–Hiliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278.
|
| [25] |
Y. Oono and S. Puri,
Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.
|
| [26] |
C. B. Schönlieb and A. Bertozzi,
Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), 413-457.
|
| [27] |
S. Villain-Guillot, Phases modulées et dynamique de Cahn–Hilliard, Habilitation thesis, Habilitation thesis, Université Bordeaux 1, 2010. |
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