Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System
Analysis of a Mixed Finite Element Method for a Cahn-Hilliard-Darcy-Stokes System
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DOI：
10.1137/130950628
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发表时间：
2013-12
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影响因子：
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通讯作者：
Amanda E. Diegel;Xiaobing H. Feng;S. Wise
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文献类型：
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作者：
Amanda E. Diegel;Xiaobing H. Feng;S. Wise
在本文中，我们设计和分析了一个修改后的Cahn-Hilliard方程耦合非稳态达西-斯托克斯流动模型相分离和耦合流体流动在不混溶的二元流体和二嵌段共聚物熔体的混合有限元方法。时间离散化是基于一个凸分裂的能量方程。我们证明了我们的计划是无条件能量稳定的空间离散模拟系统的连续自由能和无条件唯一可解。我们证明了对于任意有限的最终时刻$T$，在二维和三维空间中，相变量在$L ^\infty\left（0，T，L ^\infty\right）$中有界，化学势在$L^\infty \left（0，T，L^2\right）$中有界.随后，我们证明了这些变量收敛在适当的能量规范在二维和三维的最佳速率。
In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a non-steady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the phase variable is bounded in $L^\infty \left(0,T,L^\infty\right)$ and the chemical potential is bounded in $L^\infty \left(0,T,L^2\right)$ absolutely unconditionally in two and three dimensions, for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.