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Volume 29, Issue 4
A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations

Yaoyao Chen, Yunqing Huang & Nianyu Yi

DOI: 10.4208/cicp.OA-2020-0032

Commun. Comput. Phys., 29 (2021), pp. 1186-1212.

Published online: 2021-02

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  • Abstract

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive method is based on a linear, decoupled scheme introduced by Shen and Yang [30]. An unconditionally energy stable discrete law for the modified energy is shown for the fully discrete scheme. A superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function, respectively. Based on the proposed space and time discretization error estimators, a time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

  • Keywords

Cahn–Hilliard equation, Navier–Stokes equation, energy stability, adaptive, SCR.

  • AMS Subject Headings

65N15, 65N30, 65N50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1186, author = {Chen , YaoyaoHuang , Yunqing and Yi , Nianyu}, title = {A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations}, journal = {Communications in Computational Physics}, year = {2021}, volume = {29}, number = {4}, pages = {1186--1212}, abstract = {

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive method is based on a linear, decoupled scheme introduced by Shen and Yang [30]. An unconditionally energy stable discrete law for the modified energy is shown for the fully discrete scheme. A superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function, respectively. Based on the proposed space and time discretization error estimators, a time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0032}, url = {http://global-sci.org/intro/article_detail/cicp/18647.html} }
TY - JOUR T1 - A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations AU - Chen , Yaoyao AU - Huang , Yunqing AU - Yi , Nianyu JO - Communications in Computational Physics VL - 4 SP - 1186 EP - 1212 PY - 2021 DA - 2021/02 SN - 29 DO - http://doi.org/10.4208/cicp.OA-2020-0032 UR - https://global-sci.org/intro/article_detail/cicp/18647.html KW - Cahn–Hilliard equation, Navier–Stokes equation, energy stability, adaptive, SCR. AB -

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive method is based on a linear, decoupled scheme introduced by Shen and Yang [30]. An unconditionally energy stable discrete law for the modified energy is shown for the fully discrete scheme. A superconvergent cluster recovery based a posteriori error estimations are constructed for both the phase field variable and velocity field function, respectively. Based on the proposed space and time discretization error estimators, a time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

Chen , YaoyaoHuang , Yunqing and Yi , Nianyu. (2021). A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations. Communications in Computational Physics. 29 (4). 1186-1212. doi:10.4208/cicp.OA-2020-0032
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