Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem

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February 2016, 9(1): 269-287. doi: 10.3934/dcdss.2016.9.269

Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem

Salim Meddahi ^1, and David Mora ^2,

1.	Departamento de Matemáticas, Universidad de Oviedo, Facultad de Ciencias, Calvo Sotelo s/n, 33007 Oviedo, Spain
2.	Departamento de Matemática, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile

Received September 2014 Revised February 2015 Published December 2015

Abstract
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We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments.

Keywords: Fluid-structure interaction, error estimates., sloshing, mixed elasticity equations, eigenvalue problem, elastoacoustic vibrations, finite elements.

Mathematics Subject Classification: Primary: 65N15, 65N25, 65N30; Secondary: 74F1.

Citation: Salim Meddahi, David Mora. Nonconforming mixed finite element approximation of a fluid-structure interaction spectral problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 269-287. doi: 10.3934/dcdss.2016.9.269

References:

[1]	D. N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity, Japan J. Appl. Math., 1 (1984), 347-367. doi: 10.1007/BF03167064.
[2]	D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), 1699-1723. doi: 10.1090/S0025-5718-07-01998-9.
[3]	A. Bermúdez, R. Durán, M. A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes, SIAM J. Numer. Anal., 32 (1995), 1280-1295. doi: 10.1137/0732059.
[4]	A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg., 40 (1997), 1435-1448. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1435::AID-NME119>3.0.CO;2-P.
[5]	A. Bermúdez, P. Gamallo, L. Hervella-Nieto, R. Rodríguez and D. Santamarina, Fluid-structure Acoustic Interaction, Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods (eds. S. Marburg and B. Nolte), Springer, 2008.
[6]	A. Bermúdez and R. Rodríguez, Finite element analysis of sloshing and hydroelastic vibrations under gravity, RAIRO - Math. Model. Numer. Anal. ($M^2AN$), 33 (1999), 305-327. doi: 10.1051/m2an:1999117.
[7]	D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120. doi: 10.1017/S0962492910000012.
[8]	D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Comm. Pure Appl. Anal., 8 (2009), 95-121. doi: 10.3934/cpaa.2009.8.95.
[9]	F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.
[10]	M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer, Berlin, 1988.
[11]	J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence, RAIRO Anal. Numér., 12 (1978), 97-112.
[12]	G. N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D, SIAM J. Numer. Anal., 50 (2012), 1648-1674. doi: 10.1137/110836705.
[13]	P. Grisvard, Problèmes aux limites dans les polygones. Mode d'emploi, EDF, Bulletin de la Direction des Études et Recherches (Serie C), 1 (1986), 21-59.
[14]	R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/S0962492902000041.
[15]	L. Kiefling and G. C. Feng, Fluid-structure finite element vibration analysis, AIAA J., 14 (1976), 199-203. doi: 10.2514/3.61357.
[16]	S. Meddahi, D. Mora and R. Rodríguez, Finite element spectral analysis for the mixed formulation of the elasticity equations, SIAM J. Numer. Anal., 51 (2013), 1041-1063. doi: 10.1137/120863010.
[17]	S. Meddahi, D. Mora and R. Rodríguez, Finite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem, Comput. Math. Appl., 68 (2014), 1733-1750. doi: 10.1016/j.camwa.2014.10.016.
[18]	H. J.-P. Morand and R. Ohayon, Fluid Structure Interaction, J. Wiley & Sons, Chichester, 1995.

show all references

References:

[1]	D. N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity, Japan J. Appl. Math., 1 (1984), 347-367. doi: 10.1007/BF03167064.
[2]	D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), 1699-1723. doi: 10.1090/S0025-5718-07-01998-9.
[3]	A. Bermúdez, R. Durán, M. A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes, SIAM J. Numer. Anal., 32 (1995), 1280-1295. doi: 10.1137/0732059.
[4]	A. Bermúdez, R. Durán and R. Rodríguez, Finite element solution of incompressible fluid-structure vibration problems, Internat. J. Numer. Methods Engrg., 40 (1997), 1435-1448. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1435::AID-NME119>3.0.CO;2-P.
[5]	A. Bermúdez, P. Gamallo, L. Hervella-Nieto, R. Rodríguez and D. Santamarina, Fluid-structure Acoustic Interaction, Computational Acoustics of Noise Propagation in Fluids. Finite and Boundary Element Methods (eds. S. Marburg and B. Nolte), Springer, 2008.
[6]	A. Bermúdez and R. Rodríguez, Finite element analysis of sloshing and hydroelastic vibrations under gravity, RAIRO - Math. Model. Numer. Anal. ($M^2AN$), 33 (1999), 305-327. doi: 10.1051/m2an:1999117.
[7]	D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, 19 (2010), 1-120. doi: 10.1017/S0962492910000012.
[8]	D. Boffi, F. Brezzi and M. Fortin, Reduced symmetry elements in linear elasticity, Comm. Pure Appl. Anal., 8 (2009), 95-121. doi: 10.3934/cpaa.2009.8.95.
[9]	F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4612-3172-1.
[10]	M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, 1341, Springer, Berlin, 1988.
[11]	J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. Part 1: The problem of convergence, RAIRO Anal. Numér., 12 (1978), 97-112.
[12]	G. N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D, SIAM J. Numer. Anal., 50 (2012), 1648-1674. doi: 10.1137/110836705.
[13]	P. Grisvard, Problèmes aux limites dans les polygones. Mode d'emploi, EDF, Bulletin de la Direction des Études et Recherches (Serie C), 1 (1986), 21-59.
[14]	R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), 237-339. doi: 10.1017/S0962492902000041.
[15]	L. Kiefling and G. C. Feng, Fluid-structure finite element vibration analysis, AIAA J., 14 (1976), 199-203. doi: 10.2514/3.61357.
[16]	S. Meddahi, D. Mora and R. Rodríguez, Finite element spectral analysis for the mixed formulation of the elasticity equations, SIAM J. Numer. Anal., 51 (2013), 1041-1063. doi: 10.1137/120863010.
[17]	S. Meddahi, D. Mora and R. Rodríguez, Finite element analysis for a pressure-stress formulation of a fluid-structure interaction spectral problem, Comput. Math. Appl., 68 (2014), 1733-1750. doi: 10.1016/j.camwa.2014.10.016.
[18]	H. J.-P. Morand and R. Ohayon, Fluid Structure Interaction, J. Wiley & Sons, Chichester, 1995.

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