Error analysis of a finite element approximation of a degenerate Cahn-Hilliard equation

Agosti, A.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 827-867.

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.

Classification : 35Q80, 35K35, 35K65, 65M60, 65K10, 65G99
Mots clés : Degenerate Cahn Hilliard equation, single well potential, continuous Galerkin finite element approximation, error analysis

[1] R.A. Adams Sobolev Spaces. Academic press New York (1975).

[2] A. Agosti P.F. Antonietti P. Ciarletta M. Grasselli and M. Verani A Cahn-Hilliard type equation with degenerate mobility and single-well potential. Part I: convergence analysis of a continuous Galerkin finite element discretization. (2016).

[3] A. Agosti P.F. Antonietti P. Ciarletta M. Grasselli and M. Verani A Cahn-Hilliard type equation with application to tumor growth dynamics. To appear in: Math. Meth. Appl. Sci. DOI: (2017). | DOI | MR | Zbl

[4] S.M. Allen and J.W. Cahn Ground state structures in ordered binary alloys with second neighbor interactions. Acta Metall. 20 (1972) 423–433. | DOI

[5] J.W. Barrett and K. Deckelnick 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

[6] J.W. Barrett J.F. Blowey and H. Garcke Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. | DOI | MR | Zbl

[7] E. Beretta M. Bertsch and R. Dalpasso Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129 (1995) 175–200. | DOI | MR | Zbl

[8] F. Bernis and A. Friedman Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83 (1990) 179–206. | DOI | MR | Zbl

[9] A.L. Bertozzi S. Esedoglu and A. Gillette Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans. Image Process. 16 (2007) 285–291. | DOI | MR | Zbl

[10] H. Byrne and L. Preziosi Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20 (2003) 341–366. | DOI | Zbl

[11] J.W. Cahn On spinodal decomposition. Acta Metall. 9 (1961) 795–801. | DOI

[12] J.W. Cahn and J.E. Hilliard Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. | DOI | Zbl

[13] C. Chatelain T. Balois P. Ciarletta and M. Benamar Emergence of microstructural patterns in skin cancer: a phase separation analysis in a binary mixture. New J. Phys. 13 (2011) 115013. | DOI

[14] C. Chatelain P. Ciarletta and M.B. Amar 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

[15] M.C. Colombo C. Giverso E. Faggiano C. Boffano F. Acerbi and P. Ciarletta Towards the personalized treatment of glioblastoma: integrating patient-specific clinical data in a continuous mechanical model. PLoS One 10 (2015) e0132887. | DOI

[16] J. Erlebacher M.J. Aziz A. Karma N. Dimitrov and K. Sieradzki Evolution of nanoporosity in dealloying. Nature 410 (2001) 450–453. | DOI

[17] C.G. Gal A. Giorgini and M. Grasselli 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

[18] G. Grün and M. Rumpf Nonnegativity preserving convergent schemes for the thin film equation. Numer. Math. 87 (2000) 113–152. | DOI | MR | Zbl

[19] D. Hilhorst J. Kampmann T.N. Nguyen and K.J. Van Der Zee Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci. 25 (2014) 1011–1043. | DOI | MR | Zbl

[20] J. JostPartial Differential Equations. Springer (2007).

[21] D. Kay V. Styles and E. Süli. Discontinuous galerkin finite element approximation of the Cahn-Hilliard equation with convection. Siam J. Numer. Anal. 47 (2009) 2660–2685. | DOI | MR | Zbl

[22] E. Khain and L.M. Sander Generalized Cahn-Hilliard equation for biological applications. Phys. Rev. E 77 (2008) 051129. | DOI

[23] Q.X. Liu A. Doelman V. Rottschäfer M. De Jager P.M.J. Herman M. Rietkerk and J. Vande Koppel 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

[24] L.P. Mcmaster Aspects of liquid-liquid phase transition phenomena in multicomponent polymeric systems. Adv. Chem. Ser. 142 (1975) 43–65. | DOI

[25] J.T. Oden A. Hawkins and S. Prudhomme General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20 (2010) 477–517. | DOI | MR | Zbl

[26] M. Pierre and S. Injrou Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations. Adv. Differ. Equ. 15 (2010) 1161–1192. | MR | Zbl

[27] M. Pierre L. Cherfils and M. Petcu A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discret. Contin. Dyn. Syst. 27 (2010) 1511–1533. | DOI | MR | Zbl

[28] B. Riviere Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008). | DOI | MR | Zbl

[29] S. Tremaine On the origin of irregular structure in Saturn’s rings. Astron. J. 125 (2003) 894–901. | DOI

[30] S.M. Wise J.S. Lowengrub H.B. Frieboes and V. Cristini Three-dimensional multispecies nonlinear tumor growth I: model and numerical method. J. Theor. Biol. 253 (2008) 524–543. | DOI | MR | Zbl

[31] I. Yotov 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] Y. Zeng and M.Z. Bazant Phase separation dynamics in isotropic ion-interaction particles. SIAM J. Appl. Math. 74 (2014) 980–1004. | DOI | MR | Zbl