Analysis of the Morley element for the Cahn–Hilliard equation and the Hele-Shaw flow

Wu, Shuonan; Li, Yukun

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 1025-1052.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

The paper analyzes the Morley element method for the Cahn–Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on $\frac{1}{ϵ}$ polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on $\frac{1}{ϵ}$ polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825–847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L^∞(H²) error order, or we can merely obtain the error bounds which are exponentially dependent on $\frac{1}{ϵ}$. Numerical results are presented to validate the optimal L^∞(H²) error order and the asymptotic behavior of the solutions of the Cahn–Hilliard equation.

R.A. Adams and J.J. Fournier, Sobolev Spaces, vol. 140. Elsevier (2003). | MR | Zbl

N.D. Alikakos, P.W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal. 128 (1994) 165–205. | DOI | MR | Zbl

S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. | DOI

A.C. Aristotelous, O. Karakashian and S.M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete Continuous Dyn. Syst.-Ser. B 18 (2013) 2211–22238. | DOI | MR | Zbl

D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | DOI | MR | Zbl

S. Bartels, R. Müller and C. Ortner, Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes. SIAM J. Numer. Anal. 49 (2011) 110–134. | DOI | MR | Zbl

S.C. Brenner, Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65 (1996) 897–921. | DOI | MR | Zbl

S.C. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68 (1999) 25–53. | DOI | MR | Zbl

S.C. Brenner, Forty years of the Crouzeix-Raviart element. Numer. Methods Partial Differ. Equ. 31 (2015) 367–396. | DOI | MR

S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254 (2013) 31–42. | DOI | MR | Zbl

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

X. Chen, Spectrum for the Allen-Chan, Chan-Hillard, and phase-field equations for generic interface. Commun. Partial Differ. Equ. 19 (1994) 1371–1395. | DOI | MR | Zbl

Q. Du and R.A. Nicolaides, Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (1991) 1310–1322. | DOI | MR | Zbl

C.M. Elliott and D.A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26 (1989) 884–903. | DOI | MR | Zbl

L.C. Evans, H.M. Soner and P.E. Souganidis, Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45 (1992) 1097–1123. | DOI | MR | Zbl

X. Feng and Y. Li, Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow. IMA J. Numer. Anal. 35 (2014) 1622–1651. | DOI | MR

X. Feng and A. Prohl, Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003) 33–65. | DOI | MR | Zbl

X. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99 (2004) 47–84. | DOI | MR | Zbl

X. Feng and A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces Free Boundaries 7 (2005) 1–28. | DOI | MR | Zbl

X. Feng and H.-J. Wu, A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow. J. Comput. Math. 26 (2008) 767–796. | MR | Zbl

X. Feng, Y. Li and A. Prohl, Finite element approximations of the stochastic mean curvature flow of planar curves of graphs. Stochastic Partial Differ. Equ.: Anal. Comput. 2 (2014) 54–83. | MR | Zbl

X. Feng, Y. Li and Y. Xing, Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow. SIAM J. Numer. Anal. 54 (2016) 825–847. | DOI | MR

X. Feng, Y. Li and Y. Zhang, Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise. SIAM J. Numer. Anal. 55 (2017) 194–216. | DOI | MR

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38 (1993) 417–461. | DOI | MR | Zbl

M. Kovács, S. Larsson and A. Mesforush, Finite element approximation of the Cahn–Hilliard–Cook equation. SIAM J. Numer. Anal. 49 (2011) 2407–2429. | DOI | MR | Zbl

Y. Li, Numerical methods for deterministic and stochastic phase field models of phase transition and related geometric flows. Ph.D. thesis, The University of Tennessee (2015).

Y. Li, Error analysis of a fully discrete Morley finite element approximation for the Cahn-Hilliard equation. J. Sci. Comput. 78 (2019) 1862–1892. | DOI | MR

A. Miranville, The Cahn-Hilliard Equation: Recent Advances and Applications. SIAM, 2019. | DOI | MR

B.E. Stoth, Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Differ. Equ. 125 (1996) 154–183. | DOI | MR | Zbl

J. Xu, Y. Li, S. Wu and A. Bousquet, On the stability and accuracy of partially and fully implicit schemes for phase field modeling. Comput. Methods Appl. Mech. Eng. 345 (2019) 826–853. | DOI | MR

S. Wu and J. Xu, Multiphase Allen-Cahn and Cahn-Hilliard models and their discretizations with the effect of pairwise surface tensions. J. Comput. Phys. 343 (2017) 10–32. | DOI | MR