流体相互作用模型的粘性分离有限元方法

摘要/Abstract

摘要：

针对流体-流体相互作用模型，研究了一种全离散的粘性分离有限元方法.该方法在时间层采用了粘性分解技术和空间混合有限元方法，其中时间项包括两个步骤.第一步，采用向后Euler方法用于时间离散化，采用半隐式方法处理非线性项，并使用几何平均方法处理流体界面.然后，在第二步中，我们只解决了一个线性Stokes问题，而没有对每个单独的区域进行时间步的空间迭代.因此，粘性分离有限元方法将非线性和不可压缩性分开.此外，通过严格的分析验证了该方法的稳定性和收敛性.最后，数值实验表明了该方法的性能.

关键词: 流体相互作用模型, 粘性分离法, 稳定性, 收敛性, 有限元法

Abstract:

In this paper, a fully discrete viscosity-splitting finite element method is developed and studied for the fluid-fluid interaction model. This method applies decomposition technique of viscosity in time and mixed finite element method in space, where the temporal term includes two steps. In the first step, a backward Euler scheme is utilized for the temporal discretization, semi-implicit scheme is applied for the nonlinearity term and the geometric averaging method is used to deal with the fluid interface. Then, in the second step, we only solve a linear Stokes problem without spatial iteration per time step for each individual domain. Hence, the viscosity-splitting finite element method splits nonlinearity and incompressibility. Moreover, the stability and convergence of the method are established by rigorous analysis. Finally, numerical experiments are presented to show the performance of the proposed method.

Key words: Fluid-fluid interaction model, Viscosity-splitting method, Stability, Convergence, Finite element method

中图分类号:

O242

李伟,黄鹏展. 流体相互作用模型的粘性分离有限元方法[J]. 数学物理学报, 2020, 40(5): 1362-1380.

Wei Li,Pengzhan Huang. A Viscosity-Splitting Finite Element Method for the Fluid-Fluid Interaction Problem[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1362-1380.

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$1/ h$	1/$\Delta t$
$1/ h$	4	8	16	32	64
16	0.17219	0.16004	0.15618	0.15407	0.15309
32	0.15315	0.13811	0.13364	0.13093	0.12972
64	0.15253	0.13698	0.13237	0.12963	0.12838

$1/ h$	1/$\Delta t$
$1/ h$	4	8	16	32	64
16	0.20129	0.16255	0.15477	0.15341	0.15267
32	0.18723	0.14207	0.13183	0.13043	0.12957
64	0.18740	0.14133	0.13046	0.12907	0.12823

$1/h$	$Err({\bf u}_{1})$	Rate	$Err({\bf u}_{2})$	Rate
8	0.16393	-	0.16249	-
16	0.06295	1.38	0.06291	1.37
32	0.01817	1.79	0.01819	1.79
64	0.00565	1.68	0.00566	1.68

$1/h$	$Err({\bf u}_{1})$	Rate	$Err({\bf u}_{2})$	Rate
8	0.16246	-	0.16109	-
16	0.06252	1.32	0.06249	1.37
32	0.01812	1.79	0.01813	1.78
64	0.00568	1.67	0.00568	1.67

$1/\Delta t$	$Err({\bf u}_{1})$	Rate	$Err({\bf u}_{2})$	Rate
4	1.17735	-	1.20235	-
8	0.47077	1.32	0.48398	1.31
16	0.14117	1.74	0.14634	1.73
32	0.04148	1.77	0.04321	1.76
64	0.01380	1.59	0.01442	1.58