### Electronic Research Archive

2021, Issue 1: 1735-1752. doi: 10.3934/era.2020089
Special Issues

# Error estimates for second-order SAV finite element method to phase field crystal model

• Received: 01 March 2020 Revised: 01 July 2020 Published: 25 August 2020
• Primary, 65M12, 65M60, 35Q56

• In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order $O(\tau^2+h^2)$ in the sense of $L^2$-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

Citation: Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model[J]. Electronic Research Archive, 2021, 29(1): 1735-1752. doi: 10.3934/era.2020089

### Related Papers:

• In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order $O(\tau^2+h^2)$ in the sense of $L^2$-norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

 [1] A. J. Archer, D. J. Ratliff, A. M. Rucklidge and P. Subramanian, Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations, Phys. Rev. E, 100 (2019). doi: 10.1103/PhysRevE.100.022140 [2] Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. (2013) 51: 2851-2873. [3] S. C. Brenner, $C^0$ interior penalty methods, in Frontiers in Numerical Analysis, Lect. Notes Comput. Sci. Eng., 85, Springer, Heidelberg, 2012, 79–147. doi: 10.1007/978-3-642-23914-4_2 [4] An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. (2008) 227: 6241-6248. [5] K. R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett., 88 (2002). doi: 10.1103/PhysRevLett.88.245701 [6] K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.051605 [7] K. R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing, Phys. Rev. B, 75 (2007). doi: 10.1103/PhysRevB.75.064107 [8] D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, unpublished article, (1998), 1–15. [9] An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Engrg. (2012) 249/252: 52-61. [10] Energy stable and convergent finite element schemes for the modified phase field crystal equation. ESAIM Math. Model. Numer. Anal. (2016) 50: 1523-1560. [11] R. Guo and Y. Xu, Local discontinuous Galerkin method and high order semi-implicit scheme for the phase field crystal equation, SIAM J. Sci. Comput., 38 (2016), A105–A127. doi: 10.1137/15M1038803 [12] A high order adaptive time-stepping strategy and local discontinuous Galerkin method for the modified phase field crystal equation. Commun. Comput. Phys. (2018) 24: 123-151. [13] Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. (2009) 228: 5323-5339. [14] Numerical methods for quasicrystals. J. Comput. Phys. (2014) 256: 428-440. [15] Linear second order energy stable schemes for phase field crystal growth models with nonlocal constraints. Comput. Math. Appl. (2020) 79: 764-788. [16] An unconditionally gradient stable numerical method for the Ohta-Kawasaki model. Bull. Korean Math. Soc. (2017) 54: 145-158. [17] X. Li and J. Shen, Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Adv. Comput. Math., 46 (2020), 20pp. doi: 10.1007/s10444-020-09789-9 [18] Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation. Appl. Numer. Math. (2020) 150: 491-506. [19] A linearly second-order energy stable scheme for the phase field crystal model. Appl. Numer. Math. (2019) 140: 134-164. [20] S. Praetorius, Efficient Solvers for the Phase-Field Crystal Equation, Ph.D dissertation, Technischen Universität Dresden, 2015. [21] Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. JOM (2007) 59: 83-90. [22] Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. (2010) 28: 1669-1691. [23] The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. (2018) 353: 407-416. [24] Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. (2018) 56: 2895-2912. [25] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, 25, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-33122-0 [26] An energy-stable convex splitting for the phase-field crystal equation. Comput. Struct. (2015) 158: 355-368. [27] An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. (2011) 49: 945-969. [28] Error analysis of SAV finite element method to phase field crystal model. Numer. Math. Theor. Meth. Appl. (2020) 13: 372-399. [29] S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal. 47 (2009), 2269–2288. doi: 10.1137/080738143 [30] Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. (2006) 44: 1759-1779. [31] Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model. J. Comput. Phys. (2017) 330: 1116-1134.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Article outline

## Figures and Tables

Figures(3)  /  Tables(4)

• On This Site