向列相液晶流的模块grad-div稳定化有限元方法

摘要/Abstract

摘要：

该文针对向列相液晶流，提出了一种模块grad-div稳定化有限元方法，主要是在向后欧拉格式中增加了一个后处理步骤.该方法可以惩罚原有格式的质量不守恒性，但不会随着稳定化参数的变大而使计算时间增加.此外，该文给出了向列相液晶流的速度和分子方向的误差估计，还通过数值实验验证了理论分析.

关键词: 向列相液晶模型, 模块grad-div稳定化方法, 误差分析, 有限元方法, 向后欧拉方法

Abstract:

In this paper, we presents a modular grad-div stabilized finite element method for nematic liquid crystal flow, which adds to the backward Euler scheme a post precessing step. This method can penalize for lack of mass conservation but it does not increase computational time for increasing stabilized parameters. Moreover, error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown. Finally, the theoretical findings and numerical efficiency are verified by numerical experiments.

Key words: Nematic liquid crystal model, Modular grad-div stabilized method, Error estimates, Finite element method, Backward Euler scheme

中图分类号:

O242

李婷,黄鹏展. 向列相液晶流的模块grad-div稳定化有限元方法[J]. 数学物理学报, 2021, 41(2): 451-467.

Ting Li,Pengzhan Huang. A Modular grad-div Stabilized Finite Element Method for Nematic Liquid Crystal Flow[J]. Acta mathematica scientia,Series A, 2021, 41(2): 451-467.

图/表 4

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参考文献 35

1	Akbas M , Linke A , Rebholz L G , Schroeder P W . The analogue of grad-div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes. Comput Methods Appl Mech Engrg, 2018, 341, 917- 938 doi: 10.1016/j.cma.2018.07.019
2	Akbas M, Rebholz L G. Modular grad-div stabilization for multiphysics flow problems. 2020, arXiv: 2001. 10100
3	An R , Su J . Optimal error estimates of semi-implicit Galerkin method for time dependent nematic liquid crystal flows. J Sci Comput, 2018, 74, 979- 1008 doi: 10.1007/s10915-017-0479-7
4	Badia S , Guillén-Gónzalez F , Gutiérrez-Santacreu J V . An overview on numerical analyses of nematic liquid crystal flows. Arch Comput Methods Eng, 2011, 18, 285- 313 doi: 10.1007/s11831-011-9061-x
5	Becker R , Feng X B , Prohl A . Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow. SIAM J Numer Anal, 2008, 46, 1704- 1731 doi: 10.1137/07068254X
6	Bochev P , Dohrmann C , Gunzburger M . Stabilization of low-order mixed finite element for the Stokes equations. SIAM J Numer Anal, 2006, 44, 82- 101 doi: 10.1137/S0036142905444482
7	Brenner S , Scott L . The Mathematical Theory of Finite Element Methods. Berlin: Springer, 1994
8	Cabrales R C , Guillén-González F , Gutiérrez-Santacreu J V . A time-splitting finite element stable approximation for the Ericksen-Leslie equations. SIAM J Sci Comput, 2015, 37, B261- B282 doi: 10.1137/140960979
9	Cabrales R C , Guillén-González F , Gutiérrez-Santacreu J V . A projection-based time-splitting algorithm for approximating nematic liquid crystal flows with stretching. Z Angew Math Mech, 2017, 97, 1204- 1219 doi: 10.1002/zamm.201600247
10	Du Q , Guo B , Shen J . Fourier spectral approximation to a dissipative system modeling the flow of liquid crystals. SIAM J Numer Anal, 2001, 39, 735- 762 doi: 10.1137/S0036142900373737
11	Ericksen J . Conservation laws for liquid crystals. Trans Soc Rheol, 1961, 5, 22- 34
12	Ericksen J . Continuum theory of nematic liquid crystals. Res Mech, 1987, 21, 381- 392
13	Fiordilino J A , Layton W , Rong Y . An efficient and modular grad-div stabilization. Comput Methods Appl Mech Engrg, 2018, 335, 917- 938
14	Franca L P , Hughes T J . Two classes of mixed finite element methods. Comput Methods Appl Mech Engrg, 1988, 69, 89- 129 doi: 10.1016/0045-7825(88)90168-5
15	Girault V , Guillén-González F . Mixed formulation, approximation and decoupling algorithm for a penalized nematic liquid crystals model. Math Comp, 2011, 80, 781- 819 doi: 10.1090/S0025-5718-2010-02429-9
16	Guillén-González F , Gutiérrez-Santacreu J V . A linear mixed finite element scheme for a nematic Ericksen-Leslie liquid crystal model. ESAIM: Math Model Numer Anal, 2013, 47, 1433- 1464 doi: 10.1051/m2an/2013076
17	Guillén-González F , Koko J . A splitting in time scheme and augmented lagrangian method for a nematic liquid crystal problem. J Sci Comput, 2015, 65, 1129- 1144 doi: 10.1007/s10915-015-0002-y
18	He Y N , Wang A W , Mei L Q . Stabilized finite-element method for the stationary Navier-Stokes equations. J Engrg Math, 2005, 51, 367- 380 doi: 10.1007/s10665-004-3718-5
19	Jenkins E W , John V , Linke A , Rebholz L G . On the parameter choice in grad-div stabilization for stokes equations. Adv Comput Math, 2014, 40, 491- 516 doi: 10.1007/s10444-013-9316-1
20	Leslie F . Some constitutive equations for liquid crystals. Arch Ration Mech, 1987, 21, 381- 392
21	Lin F H . Nonlinear theory of defects in nematics liquid crystals: Phase transitation and flow phenomena. Commun Pure Appl Math, 1989, 42, 789- 814 doi: 10.1002/cpa.3160420605
22	Lin F H , Liu C . Existence of solutions for the Ericksen-Leslie system. Arch Ration Mech Anal, 2000, 154, 135- 156 doi: 10.1007/s002050000102
23	Lin F H , Lin J , Wang C . Liquid crystal flows in two dimensions. Arch Ration Mech Anal, 2010, 197, 297- 336 doi: 10.1007/s00205-009-0278-x
24	Lin F H , Wang C . On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. Chin Ann Math Ser B, 2010, 31, 921- 938 doi: 10.1007/s11401-010-0612-5
25	Linke A , Rebholz L G . On a reduced sparsity stabilization of grad-div type for incompressible flow problems. Comput Methods Appl Mech Engrg, 2013, 261/262, 142- 153 doi: 10.1016/j.cma.2013.04.005
26	Lu X , Huang P . A modular grad-div stabilization for the 2D/3D nonstationary incompressible magnetohydrodynamic equations. J Sci Comput, 2020, 82, 3 doi: 10.1007/s10915-019-01114-x
27	Minev P , Vabishchevich P N . Spliting schemes for unsteady problems involving the grad-div operator. Appl Numer Math, 2018, 124, 130- 139 doi: 10.1016/j.apnum.2017.10.005
28	Nochetto R , Pyo J H . A finite element gauge-Uzawa method. Part I: The Navier-Stokes equations. SIAM J Numer Anal, 2005, 43, 1043- 1068 doi: 10.1137/040609756
29	Olshanskii M , Reusken A . Grad-div stabilization for Stokes equations. Math Comp, 2004, 73, 1699- 1718
30	Qin Y , Hou Y , Huang P , Wang Y . Numerical analysis of two grad-div stabilization methods for the time-dependent Stokes/Darcy model. Comput Math Appl, 2020, 79, 817- 832 doi: 10.1016/j.camwa.2019.07.032
31	Rong Y, Fiordilino J A. Numerical analysis of a BDF2 modular grad-div Stabilization method for the Navier-Stokes equations. 2018, arXiv: 1806.10750
32	Song L , Hou Y , Cai Z . Recovery-based error estimator for stabilized finite element methods for the Stokes equation. Comput Meth Appl Mech Engrg, 2014, 272, 1- 16 doi: 10.1016/j.cma.2014.01.004
33	Song L , Su H , Feng X . Recovery-based error estimator for stabilized finite element method for the stationary Navier-Stokes problem. SIAM J Sci Comput, 2016, 38, A3758- A3772 doi: 10.1137/15M1015261
34	Zhang S , Liu C , Zhang H . Numerical simulations of hydrodynamics of nematic liquid crystals: Effects of kinematic transports. Commun Comput Phys, 2011, 9, 974- 993 doi: 10.4208/cicp.160110.290610a
35	Zheng H , Hou Y , Shi F . A posteriori error estimates of stabilization of low-order mixed finite elements for incompressible flow. SIAM J Sci Comput, 2010, 32, 1346- 1360 doi: 10.1137/090771508

h	$\\|\nabla\cdot({\bf u}(t_m)-{\bf u}_h^m)\\|_{0} $	收敛阶	$\\|\nabla({\bf u}(t_m)- {\bf u}_h^m)\\|_{0}$	收敛阶
1/20	0.0254707		0.0301272
1/30	0.0171152	0.9805080	0.0207659	0.9177351
1/40	0.0129294	0.9749067	0.0135101	1.4942699

h	$\\|\nabla({\bf u}(t_m)-{\bf u}_h^m)\\|_{0} $	收敛阶	$\\|\nabla({\bf d}(t_m)-{\bf d}_h^m)\\|_{0}$	收敛阶
1/30	0.0213587		0.0119277
1/40	0.0148425	1.2651621	0.0062036	2.2724003
1/50	0.0117168	1.0597257	0.0038539	2.1333538

υ	$\\|\nabla\cdot ({\bf u}(t_m)- {\bf u}_h^m)\\|_{0} $ An的算法^[3]	算法2.1	$\\|\nabla({\bf u}(t_m)- {\bf u}_h^m) \\|_{0}$ An的算法^[3]	算法2.1
1.0	0.177291	0.016364	0.155393	0.015200
1.0e-1	0.411760	0.017199	1.797790	0.049710
1.0e-2	0.098620	0.021241	5.644600	0.122195
1.0e-3	1.147560	0.022955	7.007330	0.143346
1.0e-4	1.166460	0.023173	7.177150	0.145903

β	标准grad-div	模块grad-div	增加(%)
0.1	8.811	7.578	16.271
0.2	9.877	7.461	32.381
0.4	10.418	7.488	39.129
0.8	13.185	7.437	77.289
8	F	7.566	-
80	F	7.413	-
800	F	7.458	-
8000	F	7.312	-

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