This work considers a Cahn-Hilliard type equation with degenerate mobility and single-well potential of Lennard-Jones type, motivated by increasing interest in diffuse interface modelling of solid tumors. The degeneracy set of the mobility and the singularity set of the potential do not coincide, and the zero of the potential is an unstable equilibrium configuration. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. In particular, the singularities of the potential do not compensate the degeneracy of the mobility by constraining the solution to be strictly separated from the degeneracy values. The error analysis of a well posed continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality, is developed. Whilst in previous works the error analysis of suitable finite element approximations has been studied for second order degenerate and fourth order non degenerate parabolic equations, in this work the a priori estimates of the error between the discrete solution and the weak solution to which it converges are obtained for a degenerate fourth order parabolic equation. The theoretical error estimates obtained in the present case state that the norms of the approximation errors, calculated on the support of the solution in the proper functional spaces, are bounded by power laws of the discretization parameters with exponent 1/2, while in the case of the classical Cahn-Hilliard equation with constant mobility the exponent is 1. The estimates are finally succesfully validated by simulation results in one and two space dimensions.
Mots-clés : Degenerate Cahn Hilliard equation, single well potential, continuous Galerkin finite element approximation, error analysis
@article{M2AN_2018__52_3_827_0, author = {Agosti, A.}, title = {Error analysis of a finite element approximation of a degenerate {Cahn-Hilliard} equation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {827--867}, publisher = {EDP-Sciences}, volume = {52}, number = {3}, year = {2018}, doi = {10.1051/m2an/2018018}, mrnumber = {3865551}, zbl = {1405.35221}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2018018/} }
TY - JOUR AU - Agosti, A. TI - Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 827 EP - 867 VL - 52 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2018018/ DO - 10.1051/m2an/2018018 LA - en ID - M2AN_2018__52_3_827_0 ER -
%0 Journal Article %A Agosti, A. %T Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 827-867 %V 52 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2018018/ %R 10.1051/m2an/2018018 %G en %F M2AN_2018__52_3_827_0
Agosti, A. Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 827-867. doi : 10.1051/m2an/2018018. http://archive.numdam.org/articles/10.1051/m2an/2018018/
[1] Sobolev Spaces. Academic press New York (1975).
[2] A Cahn-Hilliard type equation with degenerate mobility and single-well potential. Part I: convergence analysis of a continuous Galerkin finite element discretization. (2016).
and[3] A Cahn-Hilliard type equation with application to tumor growth dynamics. To appear in: Math. Meth. Appl. Sci. DOI: (2017). | DOI | MR | Zbl
and[4] Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metall. 20 (1972) 423–433. | DOI
and[5] Existence uniqueness and approximation of a doubly-degenerate nonlinear parabolic system modelling bacterial evolution. Math. Models Methods Appl. Sci. 17 (2007) 1095–1127. | DOI | MR | Zbl
and[6] Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. | DOI | MR | Zbl
and[7] Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129 (1995) 175–200. | DOI | MR | Zbl
and[8] Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83 (1990) 179–206. | DOI | MR | Zbl
and[9] Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans. Image Process. 16 (2007) 285–291. | DOI | MR | Zbl
and[10] Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20 (2003) 341–366. | DOI | Zbl
and[11] On spinodal decomposition. Acta Metall. 9 (1961) 795–801. | DOI
[12] Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. | DOI | Zbl
and[13] Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture. New J. Phys. 13 (2011) 115013. | DOI
and[14] Morphological changes in early melanoma development: influence of nutrients growth inhibitors and cell-adhesion mechanisms. J. Theor. Biol. 290 (2011) 46–59. | DOI | MR | Zbl
and[15] Towards the personalized treatment of glioblastoma: integrating patient-specific clinical data in a continuous mechanical model. PLoS One 10 (2015) e0132887. | DOI
and[16] Evolution of nanoporosity in dealloying. Nature 410 (2001) 450–453. | DOI
and[17] The nonlocal Cahn-Hilliard equation with singular potential: well-posedness regularity and strict separation property. J. Differ. Equ. 263 (2017) 5253–5297. | DOI | MR | Zbl
and[18] Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113–152. | DOI | MR | Zbl
and[19] Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25 (2014) 1011–1043. | DOI | MR | Zbl
and[20] Partial Differential Equations. Springer (2007).
[21] Discontinuous galerkin finite element approximation of the Cahn-Hilliard equation with convection. Siam J. Numer. Anal. 47 (2009) 2660–2685. | DOI | MR | Zbl
and[22] Generalized Cahn-Hilliard equation for biological applications. Phys. Rev. E 77 (2008) 051129. | DOI
and[23] Phase separation explains a new class of self-organized spatial patterns in ecological systems. Proc. Natl. Acad. Sci. USA 110 (2013) 11905–11910. | DOI | MR
and[24] Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems. Adv. Chem. Ser. 142 (1975) 43–65. | DOI
[25] General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20 (2010) 477–517. | DOI | MR | Zbl
and[26] Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations. Adv. Differ. Equ. 15 (2010) 1161–1192. | MR | Zbl
and[27] A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discret. Contin. Dyn. Syst. 27 (2010) 1511–1533. | DOI | MR | Zbl
and[28] Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008). | DOI | MR | Zbl
[29] On the origin of irregular structure in Saturn’s rings. Astron. J. 125 (2003) 894–901. | DOI
[30] Three-dimensional multispecies nonlinear tumor growth I: model and numerical method. J. Theor. Biol. 253 (2008) 524–543. | DOI | MR | Zbl
and[31] A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow. J. Numer. Math. 5 (1997) 211–230. | MR | Zbl
[32] Phase separation dynamics in isotropic ion-interaction particles. SIAM J. Appl. Math. 74 (2014) 980–1004. | DOI | MR | Zbl
andCité par Sources :