Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem

Xin Zhao; Xin Liu; Jian Li; Xin Zhao; Xin Liu; Jian Li

doi:10.3934/math.2020337

AIMS Mathematics

2020, Volume 5, Issue 5: 5240-5260. doi: 10.3934/math.2020337

Previous Article Next Article

Research article

Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem

Xin Zhao ^{1
,
,},
Xin Liu ^{2
,
,},
Jian Li ^{3
,
,}

1.
Department of Geographical Science and Environment Engineering, Key Laboratory of Disaster Monitoring and Mechanism Simulation of Shaanxi Province, Baoji University of Arts and Sciences, Baoji 721013, P. R. China
2.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710129, P. R. China
3.
Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China

Received: 25 April 2020 Accepted: 08 June 2020 Published: 18 June 2020
MSC : Primary 35J20; Secondary 65N08, 76D05

In this paper, a semi-discrete finite element method for the nonlinear fluid-structure interaction problem interacts between the Navier-Stokes fluids and linear elastic solids, is studied and developed. A classical mixed variational principle of the weak formulation is given, and the corresponding finite element method is defined. As for the nonlinearity arising from the nonlinear interaction problem, we consider in time of a solution for suitably small data, and uniqueness hypothesis. This approach is fairly robust and adapts to the important case of interface with fractures or cracks. Convergence and estimate of the finite element method are also obtained for the nonlinear fluid-structure interaction problem. Finally, numerical experiments are presented to show the performance of the proposed method.
- fluid-structure interaction,
- Navier-Stokes equations,
- finite element method,
- convergence analysis,
- error estimate,
- numerical analysis
Citation: Xin Zhao, Xin Liu, Jian Li. Convergence analysis and error estimate of finite element method of a nonlinear fluid-structure interaction problem[J]. AIMS Mathematics, 2020, 5(5): 5240-5260. doi: 10.3934/math.2020337

Related Papers:

Abstract

In this paper, a semi-discrete finite element method for the nonlinear fluid-structure interaction problem interacts between the Navier-Stokes fluids and linear elastic solids, is studied and developed. A classical mixed variational principle of the weak formulation is given, and the corresponding finite element method is defined. As for the nonlinearity arising from the nonlinear interaction problem, we consider in time of a solution for suitably small data, and uniqueness hypothesis. This approach is fairly robust and adapts to the important case of interface with fractures or cracks. Convergence and estimate of the finite element method are also obtained for the nonlinear fluid-structure interaction problem. Finally, numerical experiments are presented to show the performance of the proposed method.

References

[1]	R. A. Adams, Sobolev Spaces, Academic press, New York, 1975.
[2]	M. Astorino and C. Grandmont, Convergence analysis of a projection semi-implicit coupling scheme for fluid-structure interaction problems, Numer. Math., 116 (2010), 721-767. doi: 10.1007/s00211-010-0311-x
[3]	D. Boffi, L. Gastaldi, A fictitious domain approach with Lagrange multiplier for fluid-structure interactions, Numer. Math., 135 (2017), 711-732. doi: 10.1007/s00211-016-0814-1
[4]	J. Boujot, Mathematical formulation of fluid-structure interaction problems, RAIRO Model. Math. Anal. Numer., 21 (1987), 239-260. doi: 10.1051/m2an/1987210202391
[5]	T. Chacon-Rebollo, V. Girault, F. Murat, et al. Analysis of a Coupled Fluid-Structure Model with Applications to Hemodynamics, SIAM J. Numer. Anal., 54 (2016), 994-1019. doi: 10.1137/140991509
[6]	Z. Chen, Finite Element Methods and Their Applications, Spring-Verlag, Heidelberg, 2005.
[7]	P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
[8]	Q. Du, M. Gunzburger, L. Hou, et al. Analysis of a linear fluid-structure interaction problem, Disc. Cont. Dyn. Syst., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633
[9]	Q. Du, M. Gunzburger, L. Hou, et al. Semidiscrete finite element approximations of a linear fluidstructure interaction problem, SIAM J. Nmer. Anal., 42 (2004), 1-29. doi: 10.1137/S0036142903408654
[10]	M. Fernández, Incremental displacement-correction schemes for incompressible fluid-structure interaction, Numer. Math., 123 (2013), 21-65. doi: 10.1007/s00211-012-0481-9
[11]	F. Flori and P. Orenga, Fluid-structure interaction: Analysis of a 3-D compressible model, Ann. Inst. H. Poincare Anal. Non Lineaire, 17 (2000), 753-777. doi: 10.1016/S0294-1449(00)00119-0
[12]	V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1987.
[13]	C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem, M2AN Math. Model. Numer. Anal., 34 (2000), 609-636. doi: 10.1051/m2an:2000159
[14]	Y. He, Y. Lin, W. Sun, Stabilized finite element method for the non-stationary Navier-Stokes problem, Disc. Cont. Dyn. Syst. Series B, 6 (2006), 41-68.
[15]	F. Hecht and O. Pironneau, An Energy Stable Monolithic Eulerian Fluid-Structure Finite Element Method, Int. J. Numer. Meth. Fl., 85 (2017), 430-446. doi: 10.1002/fld.4388
[16]	J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018
[17]	J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary NavierStokes problem. Part IV: Error estimates for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. doi: 10.1137/0727022
[18]	A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20 (2000), 633-667. doi: 10.1093/imanum/20.4.633
[19]	G. Hou, J. Wang, A. Layton, Numerical methods for fluid-structure interaction-A review, Commun. Comput. Phys., 12 (2012), 337-377. doi: 10.4208/cicp.291210.290411s
[20]	G. Hsiao, R. Kleinman and G. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr., 218 (2000), 139-163. doi: 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S
[21]	R. A. Khurram and A. Masud, A multiscale/stabilized formulation of the incompressible NavierStokes equations for moving boundary flows and fluid-structure interaction, Comput. Mech., 38 (2006), 403-416. doi: 10.1007/s00466-006-0059-4
[22]	W. Layton, Introduction to the Numerical Analysis of Incompressible Viscous Flows, Comput. Sci. Eng., SIAM, Philadelphia, 2008.
[23]	P. LeTallec and S. Mani, Numerical analysis of a linearized fluid-structure interaction problem, Numer. Math., 87 (2000), 317-354. doi: 10.1007/s002110000183
[24]	J. Li, Y. He and Z. Chen, A new stabilized finite element method for the transient Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 197 (2009), 22-35.
[25]	S. Micu and E. Zuazua, Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise, SIAM J. Math. Anal.,29 (1998), 967-1001. doi: 10.1137/S0036141096312349
[26]	H. Morand and R. Ohayon, Fluid Structure Interaction: Applied Numerical Methods, Academic press, New York, 1975.
[27]	T. Richter, A fully Eulerian formulation for fluid structure interaction problems, J. Comput. Phys., 233 (2013), 227-240. doi: 10.1016/j.jcp.2012.08.047
[28]	R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Third ed., North-Holland, Amsterdam, 1984.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)