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

Liupeng Wang; Yunqing Huang; Liupeng Wang; Yunqing Huang

doi:10.3934/era.2020089

Electronic Research Archive

2021, Volume 29, Issue 1: 1735-1752. doi: 10.3934/era.2020089

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

Liupeng Wang ^{1
,
,},
Yunqing Huang ^{2
,}

1.
School of Computational Science and Electronics, Hunan Institute of Engineering, Xiangtan 411104, Hunan, China
2.
School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and, Engineering, Xiangtan University, Xiangtan 411105, Hunan, China

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.
- Error estimates,
- energy stability,
- finite element method,
- scalar auxiliary variable approach,
- Crank-Nicolson scheme,
- phase field crystal model
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

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Abstract

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.

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