流体相互作用模型的粘性分离有限元方法

Abstract

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

CLC Number:

O242

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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References 35

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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

A Viscosity-Splitting Finite Element Method for the Fluid-Fluid Interaction Problem

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$1/h$	$Err({\bf u}_{1})$	Rate	$Err({\bf u}_{2})$	Rate
8	0.14839	-	0.10273	-
16	0.05853	1.34	0.03480	1.56
32	0.01778	1.72	0.01176	1.56
64	0.00612	1.54	0.00504	1.22

$1/\Delta t$	$Err({\bf u}_{1})$	Rate	$Err({\bf u}_{2})$	Rate
4	1.17745	-	0.73266	-
8	0.47079	1.32	0.26835	1.45
16	0.14089	1.74	0.08674	1.63
32	0.04143	1.77	0.02884	1.59
64	0.01401	1.56	0.01194	1.27