A second-order numerical scheme for the Ericksen-Leslie equation

Danxia Wang; Ni Miao; Jing Liu; Danxia Wang; Ni Miao; Jing Liu

doi:10.3934/math.2022867

AIMS Mathematics

2022, Volume 7, Issue 9: 15834-15853. doi: 10.3934/math.2022867

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A second-order numerical scheme for the Ericksen-Leslie equation

College of Mathematics, Taiyuan University of Technology, Yuci, Jinzhong, 030600, Shanxi, China

Received: 29 March 2022 Revised: 19 June 2022 Accepted: 22 June 2022 Published: 27 June 2022
MSC : 35G46

In this paper, we consider a finite element approximation for the Ericksen-Leslie model of nematic liquid crystal. Based on a saddle-point formulation of the director vector, a second-order backward differentiation formula (BDF) numerical scheme is proposed, where a pressure-correction strategy is used to decouple the computation of the pressure from that of the velocity. Designing this scheme leads to solving a linear system at each time step. Furthermore, via implementing rigorous theoretical analysis, we prove that the proposed scheme enjoys the energy dissipation law. Some numerical simulations are also performed to demonstrate the accuracy of the proposed scheme.
- Ericksen-Leslie equation,
- nematic liquid crystal,
- finite element,
- second-order accuracy,
- pressure-correction
Citation: Danxia Wang, Ni Miao, Jing Liu. A second-order numerical scheme for the Ericksen-Leslie equation[J]. AIMS Mathematics, 2022, 7(9): 15834-15853. doi: 10.3934/math.2022867

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Abstract

In this paper, we consider a finite element approximation for the Ericksen-Leslie model of nematic liquid crystal. Based on a saddle-point formulation of the director vector, a second-order backward differentiation formula (BDF) numerical scheme is proposed, where a pressure-correction strategy is used to decouple the computation of the pressure from that of the velocity. Designing this scheme leads to solving a linear system at each time step. Furthermore, via implementing rigorous theoretical analysis, we prove that the proposed scheme enjoys the energy dissipation law. Some numerical simulations are also performed to demonstrate the accuracy of the proposed scheme.

References

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