Energy stable and convergent finite element schemes for the modified phase field crystal equation
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1523-1560.

We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and on a Galerkin approximation in H 1 for the phase function. This formulation includes the classical continuous finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn–Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. This is the first proof of convergence for the scheme of Gomez and Hughes, which has been shown to be unconditionally energy stable for several Cahn–Hilliard related equations. Using a Łojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations with continuous piecewise linear (P 1 ) finite elements illustrate the theoretical results.

Received:
Accepted:
DOI: 10.1051/m2an/2015092
Classification: 65M60, 65P40, 74N05, 82C26
Keywords: Finite elements, second-order schemes, gradient-like systems, Łojasiewicz inequality
Grasselli, Maurizio 1; Pierre, Morgan 2

1 Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy.
2 Laboratoire de Mathématiques et Applications UMR CNRS 7348, Université de Poitiers, Téléport 2 - BP 30179, boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France.
@article{M2AN_2016__50_5_1523_0,
     author = {Grasselli, Maurizio and Pierre, Morgan},
     title = {Energy stable and convergent finite element schemes for the modified phase field crystal equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1523--1560},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {5},
     year = {2016},
     doi = {10.1051/m2an/2015092},
     mrnumber = {3554551},
     zbl = {1358.82025},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015092/}
}
TY  - JOUR
AU  - Grasselli, Maurizio
AU  - Pierre, Morgan
TI  - Energy stable and convergent finite element schemes for the modified phase field crystal equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 1523
EP  - 1560
VL  - 50
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015092/
DO  - 10.1051/m2an/2015092
LA  - en
ID  - M2AN_2016__50_5_1523_0
ER  - 
%0 Journal Article
%A Grasselli, Maurizio
%A Pierre, Morgan
%T Energy stable and convergent finite element schemes for the modified phase field crystal equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 1523-1560
%V 50
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015092/
%R 10.1051/m2an/2015092
%G en
%F M2AN_2016__50_5_1523_0
Grasselli, Maurizio; Pierre, Morgan. Energy stable and convergent finite element schemes for the modified phase field crystal equation. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1523-1560. doi : 10.1051/m2an/2015092. http://www.numdam.org/articles/10.1051/m2an/2015092/

A. Abourou Ella and A. Rougirel, Steady state bifurcations for phase field crystal equations with underlying two dimensional kernel. Electron. J. Qual. Theory Differ. Equ. 60 (2015) 1–36. | DOI | MR | Zbl

N.E. Alaa and M. Pierre, Convergence to equilibrium for discretized gradient-like systems with analytic features. IMA J. Numer. Anal. 33 (2013) 1291–1321. | DOI | MR | Zbl

R. Backofen, A. Rätz and A. Voigt, Nucleation and growth by a phase field crystal (PFC) model. Philos. Mag. Lett. 87 (2007) 813–820. | DOI

A. Baskaran, Z. Hu, J. S. Lowengrub, C. Wang, S.M. Wise and P. Zhou, Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation. J. Comput. Phys. 250 (2013) 270–292. | DOI | MR | Zbl

A. Baskaran, J. S. Lowengrub, C. Wang and S.M. Wise, Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51 (2013) 2851–2873. | DOI | MR | Zbl

J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Amer. Math. Soc. 362 (2010) 3319–3363. | DOI | MR | Zbl

M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. 228 (2009) 5323–5339. | MR | Zbl

K.R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystal. Phys. Rev. E 90 (2004) 051605. | DOI

K.R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth. Phys. Rev. Lett. 88 (2002) 245701. | DOI

C.M. Elliott, The Cahn–Hilliard model for the kinetics of phase separation, in Mathematical models for phase change problems (Òbidos 1988). Birkhäuser, Basel (1989) 35–73. | MR | Zbl

M. Elsey and B. Wirth, A simple and efficient scheme for phase field crystal simulation. ESAIM: M2AN 47 (2013) 1413–1432. | DOI | Numdam | MR | Zbl

H. Emmerich, L. Gránásy and H. Löven, Selected issues of phase-field crystal simulations. Eur. Phys. J. Plus 126 (2011) 102. | DOI

H. Emmerich, H. Löwen, R. Wittkowskib, T. Gruhna, G. I. Tóth, G. Tegze and L. Gránásy, Phase field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Adv. Phys. 61 (2012) 665–743. | DOI

E. Feireisl and F. Simondon, Convergence for semilinear degenerate parabolic equations in several space dimensions. J. Dyn. Differ. Equ. 12 (2000) 647–673. | DOI | MR | Zbl

P. Galenko and K. Elder, Marginal stability analysis of the phase field crystal model in one spatial dimension. Phys. Rev. B 83 (2011) 064113. | DOI

P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems. Phys. Rev. E 71 (2005) 046125. | DOI

P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics. Phys. Rev. E 79 (2009) 051110. | DOI | MR

P.K. Galenko, H. Gomez, N.V. Kropotin and K.R. Elder, Unconditionally stable and numerical solution of the hyperbolic phase-field crystal equation. Phys. Rev. E 88 (2013) 013310. | DOI

H. Gomez and T.J.R. Hughes, Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models. J. Comput. Phys. 230 (2011) 5310–5327. | DOI | MR | Zbl

H. Gomez and X. Nogueira, An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Engrg. 249/252 (2012) 52–61. | DOI | MR | Zbl

M. Grasselli and M. Pierre, A splitting method for the Cahn–Hilliard equation with inertial term. Math. Models Methods Appl. Sci. 20 (2010) 1–28. | DOI | MR | Zbl

M. Grasselli and M. Pierre, Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Commun. Pure Appl. Anal. 11 (2012) 2393–2416. | DOI | MR | Zbl

M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation. Math. Models Methods Appl. Sci. 24 (2014) 2743–2783. | DOI | MR | Zbl

M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation. Discrete Contin. Dyn. Syst. Ser. A 35 (2015) 2539–2564. | DOI | MR | Zbl

M. Grasselli, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn–Hilliard equation with inertial term. J. Evol. Equ. 9 (2009) 371–404. | DOI | MR | Zbl

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn–Hilliard equation with inertial term. Comm. Partial Differential Equ. 34 (2009) 137–170. | DOI | MR | Zbl

M. Grasselli, G. Schimperna and S. Zelik, Trajectory and smooth attractors for Cahn–Hilliard equations with inertial term. Nonlin. 23 (2010) 707–737. | DOI | MR | Zbl

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl

Z. Hu, S. Wise, C. Wang and J. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the phase-field crystal equation. J. Comput. Phys. 228 (2009) 5323–5339. | DOI | MR | Zbl

S.-Z. Huang and P. Takáč, Convergence in gradient-like systems which are asymptotically autonomous and analytic. Nonlin. Anal. 46 (2001) 675–698. | DOI | MR | Zbl

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris (1969). | MR | Zbl

S. Łojasiewicz, Ensembles semi-analytiques. I.H.E.S Lecture note, Ecole Polytechnique, Paris (1965).

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications. Commun. Pure Appl. Anal. 9 (2010) 685–702. | DOI | MR | Zbl

H. Ohnogi and Y. Shiwa, Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation. Physica D. 237 (2008) 3046–3052. | DOI | MR | Zbl

M. Pierre and A. Rougirel, Stationary solutions to phase field crystal equations. Math. Methods Appl. Sci. 34 (2011) 278–308. | DOI | MR | Zbl

N. Provatas, J.A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K.R. Elder, Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. J. Miner. Metals Mater. Soc. 59 (2007) 83–90. | DOI

J. Simon, Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystals with elastic interactions. Phys. Rev. Lett. 96 (2006) 225504. | DOI

P. Stefanovic, M. Haataja and N. Provatas, Phase-field crystal study of deformation and plasticity in nanocrystalline materials. Phys. Rev. E 80 (2009) 046107. | DOI

J. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15 (1977) 317–328. | DOI

C. Wang and S.M. Wise, Global smooth solutions of the three-dimensional modified phase field crystal equation. Methods Appl. Anal. 17 (2010) 191–211. | DOI | MR | Zbl

C. Wang and S.M. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49 (2011) 945–969. | DOI | MR | Zbl

S. Wise, C. Wang and J. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47 (2009) 2269–2288. | DOI | MR | Zbl

X. Wu, G.J. Van Zwieten and K.G. Van Der Zee, Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed. Eng. 30 (2014) 180–203. | DOI | MR

Cited by Sources: