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.

参考文献

1	Aggul M , Connors J M , Erkmen D , Labovsky A E . A defect-deferred correction method for fluid-fluid interaction. SIAM J Numer Anal, 2018, 56 (4): 2484- 2512
2	Arnold D N , Brezzi F , Fortin M . A stable finite element for the Stokes equations. Calcolo, 1984, 21 (4): 337- 344
3	Bernardi C , Chacon T , Lewandowski R , Murat F . A model for two coupled turbulent fluids Ⅱ:Numerical analysis of a spectral discretization. SIAM J Numer Anal, 2003, 40 (6): 2368- 2394
4	Bernardi C , Rebello T C , Mármol M G , et al. A model for two coupled turbulent fluids Ⅲ:Numerical approximation by finite elements. Numer Math, 2004, 98 (1): 33- 66
5	Blasco J , Codina R . Error estimates for a viscosity-splitting finite element method for the incompressible Navier-Stokes equations. Appl Numer Math, 2004, 51, 1- 17
6	Blasco J , Codina R , Huerta A . A fractional-step method for the incompressible Navier-Stokes equations related to a predictor-multicorrector algorithm. Int J Numer Methods Fluids, 1998, 28 (10): 1391- 1419
7	Bresch D , Koko J . Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids. Int J Appl Math Comput Sci, 2006, 16 (4): 419- 429
8	Chorin A J . Numerical solution of the Navier-Stokes equations. Math Comput, 1968, 22, 745- 762
9	Connors J M . An ensemble-based conventional turbulence model for fluid-fluid interaction. Inter J Numer Anal Model, 2018, 15 (4): 492- 519
10	Connors J M , Ganis B . Stability of algorithms for a two domain natural convection problem and observed model uncertainty. Comput Geosci, 2011, 15 (3): 509- 527
11	Connors J M , Howell J S . A fluid-fluid interaction method using decoupled subproblems and differing time steps. Numer Meth Part Differ Equ, 2012, 28 (4): 1283- 1308
12	Connors J M , Howell J S , Layton W J . Partitioned time stepping for a parabolic two domain problem. SIAM J Numer Anal, 2009, 47 (5): 3526- 3549
13	Connors J M , Howell J S , Layton W J . Decoupled time stepping methods for fluid-fluid interaction. SIAM J Numer Anal, 2012, 50 (3): 1297- 1319
14	Fernández M A , Gerbeau J , Smaldone S . Explicit coupling schemes for a fluid-fluid interaction problem arising in hemodynamics. SIAM J Sci Comput, 2014, 36 (6): A2557- A2583
15	Girault V , Rivíere B , Wheeler M F . A splitting method using discontinuous Galerkin for the transient incompressible Navier-Stokes equations. ESAIM:Math Model Numer Anal, 2005, 39 (6): 1115- 1147
16	Guermond J L , Salgado A . Error analysis of a fractional time-stepping technique for incompressible flows with variable density. SIAM J Numer Anal, 2011, 49 (3): 917- 944
17	Guillén-González F , Redondo-Neble M V . New error estimates for a viscosity splitting scheme in time for the three-dimensional Navier-Stokes equations. IMA J Numer Anal, 2011, 31 (2): 556- 579
18	Guillén-González F , Redondo-Neble M V . Spatial error estimates for a finite element viscosity-splitting scheme for the Navier-Stokes equations. Inter J Numer Anal Model, 2013, 10 (4): 826- 844
19	Guillén-González F , Redondo-Neble M V . Convergence and error estimates of viscosity-splitting finite-element schemes for the primitive equations. Appl Numer Math, 2017, 111, 219- 245
20	Heywood J , Rannacher R . Finite element approximation of the nonstationary Navier-Stokes equations, Ⅳ:Error analysis for second order time discretizations. SIAM J Numer Anal, 1990, 27 (2): 353- 384
21	Li J , Huang P , Zhang C , Guo G . A linear, decoupled fractional time-stepping method for the nonlinear fluid-fluid interaction. Numer Methods Part Differ Equa, 2019, 35, 1873- 1889
22	Li J , Huang P , Su J , Chen Z . A linear, stabilized, non-spatial iterative, partitioned time stepping method for the nonlinear Navier-Stokes/Navier-Stokes interaction model. Bound Value Probl, 2019
23	Lions J L , Temam R , Wang S . Models for the coupled atmosphere and ocean (CAO Ⅰ). Comput Mech Adv, 1993, 1, 5- 54
24	Lions J L , Temam R , Wang S . Numerical analysis of the coupled atmosphere-ocean models (CAO Ⅱ). Comput Mech Adv, 1993, 1, 55- 119
25	Lions J L , Temam R , Wang S . Mathematical theory for the coupled atmosphere-ocean models (CAO Ⅲ). J Math Pures Appl, 1995, 74, 105- 163
26	Qian L Z , Chen J R , Feng X L . Local projection stabilized and characteristic decoupled scheme for the fluid-fluid interaction problems. Numer Meth Part Differ Equa, 2017, 33 (3): 704- 723
27	Quarteroni A , Saleri F , Veneziani A . Factorization methods for the numerical approximation of Navier-Stokes equations. Comput Methods Appl Mech Eng, 2000, 188 (1-3): 505- 526
28	Saleri F , Veneziani A . Pressure correction algebraic splitting methods for the incompressible Navier-Stokes equations. SIAM J Numer Anal, 2005, 43 (1): 174- 194
29	Strikwerda J C , Young S L . The accuracy of the fractional step method. SIAM J Numer Anal, 1999, 37 (1): 37- 47
30	Temam R. Navier-Stokes Equations, Theory and Numerical Analysis. Amsterdam: North Holland, 1984
31	Zhang Y H , Hou Y R , Shan L . Stability and convergence analysis of a decoupled algorithm for a fluid-fluid interaction problem. SIAM J Numer Anal, 2016, 54 (5): 2833- 2867
32	Zhang Y H , Hou Y R , Shan L . Error estimates of a decoupled algorithm for a fluid-fluid interaction problem. J Comput Appl Math, 2018, 333, 266- 291
33	Zhang T , Pedro D , Yuan J Y . A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem. Adv Comput Math, 2015, 41 (1): 149- 190
34	Zhang T , Qian Y X . The time viscosity-splitting method for the Boussinesq problem. J Math Anal Appl, 2017, 445 (1): 186- 211
35	Zhang T , Qian Y X , Yuan J Y . The fully discrete fractional-step method for the Oldroyd model. Appl Numer Math, 2018, 129, 83- 103

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