A decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system

Haifeng Zhang; Danxia Wang; Zhili Wang; Hongen Jia; Haifeng Zhang; Danxia Wang; Zhili Wang; Hongen Jia

doi:10.3934/math.2021505

AIMS Mathematics

2021, Volume 6, Issue 8: 8681-8704. doi: 10.3934/math.2021505

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Research article

A decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system

College of Mathematics, Taiyuan University of Technology, 030024, Tai'yuan, China

Received: 22 January 2021 Accepted: 24 May 2021 Published: 08 June 2021
MSC : 35Q30, 74S05

In this paper, a decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system with double well potential is constructed. The proposed scheme is based on the convex splitting of the energy functional in time and uses the mixed finite element discretzation in space. The computation of the velocity $ \mathbf{u} $ is separated from the computation of the pressure $ p $ by using an operator-splitting strategy. A Possion equation is solved to update the pressure at each time step. Unconditional stability and error estimates are analyzed in detail theoretically. Furthermore, the theoretical part is verified by several numerical examples, whose results show that the numerical examples are consistent with the results of the theoretical part.
- decoupled,
- modified Cahn-Hilliard-Hele-Shaw system,
- double well potential,
- convex splitting
Citation: Haifeng Zhang, Danxia Wang, Zhili Wang, Hongen Jia. A decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system[J]. AIMS Mathematics, 2021, 6(8): 8681-8704. doi: 10.3934/math.2021505

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Abstract

In this paper, a decoupled finite element method for a modified Cahn-Hilliard-Hele-Shaw system with double well potential is constructed. The proposed scheme is based on the convex splitting of the energy functional in time and uses the mixed finite element discretzation in space. The computation of the velocity $ \mathbf{u} $ is separated from the computation of the pressure $ p $ by using an operator-splitting strategy. A Possion equation is solved to update the pressure at each time step. Unconditional stability and error estimates are analyzed in detail theoretically. Furthermore, the theoretical part is verified by several numerical examples, whose results show that the numerical examples are consistent with the results of the theoretical part.

References

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