# American Institute of Mathematical Sciences

December  2016, 5(4): 515-531. doi: 10.3934/eect.2016017

## A uniform discrete inf-sup inequality for finite element hydro-elastic models

 1 Department of Mathematics, University of Nebraska, Lincoln, Lincoln, NE, 68588, United States 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received  June 2016 Revised  August 2016 Published  October 2016

A seminal result concerning finite element (FEM) approximations of the Stokes equation was the discrete inf-sup inequality that is uniform with respect to the mesh size parameter. This inequality leads to optimal error estimates for the FEM scheme. The original version pertains to the Stokes system with non-slip boundary condition on the entire boundary. On the other hand, in fluid-structure interaction problems, the interface dynamics between the fluid and the solid satisfies velocity and stress matching constraints. As a result, the pressure variable is no longer determined up to a constant and becomes subject to non-homogeneous Dirichlet conditions on the common interface. In this context, we establish a uniform discrete inf-sup estimate for a fluid-structure FEM implementation based on Taylor-Hood elements, and use this inequality to verify some stability and error estimates of this numerical scheme. An added benefit of this framework is that it does not require the Poisson-equation approach to solve for the pressure variable.
Citation: George Avalos, Daniel Toundykov. A uniform discrete inf-sup inequality for finite element hydro-elastic models. Evolution Equations & Control Theory, 2016, 5 (4) : 515-531. doi: 10.3934/eect.2016017
##### References:
 [1] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, in Fluids and waves, (2007), 15. doi: 10.1090/conm/440/08475. [2] G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system,, J. Differential Equations, 258 (2015), 4398. doi: 10.1016/j.jde.2015.01.037. [3] G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method,, Appl. Math. (Warsaw), 35 (2008), 259. doi: 10.4064/am35-3-2. [4] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. [5] O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, vol. 35 of Classics in Applied Mathematics,, Society for Industrial and Applied Mathematics (SIAM), (2001). doi: 10.1137/1.9780898719253. [6] I. Babuška, Error-bounds for finite element method,, Numer. Math., 16 (): 322. [7] I. Babuška and V. Nistor, Interior numerical approximation of boundary value problems with a distributional data,, Numer. Methods Partial Differential Equations, 22 (2006), 79. doi: 10.1002/num.20086. [8] I. Babuška, V. Nistor and N. Tarfulea, Approximate and low regularity Dirichlet boundary conditions in the generalized finite element method,, Math. Models Methods Appl. Sci., 17 (2007), 2115. doi: 10.1142/S0218202507002571. [9] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55. doi: 10.1090/conm/440/08476. [10] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables,, Numer. Math., 33 (1979), 211. doi: 10.1007/BF01399555. [11] L. Bociu, S. Derochers and D. Toundykov, Linearized hydroelasticity: A numerical study,, To appear in: Evol. Equ. Control Theory, (2016). [12] L. Bociu, D. Toundykov and J.-P. Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction,, SIAM J. Math. Anal., 47 (2015), 1958. doi: 10.1137/140970689. [13] L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid/nonlinear elasticity interaction,, Discrete Contin. Dyn. Syst., 1 (2011), 184. [14] L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid,, in Modern aspects of the theory of partial differential equations, (2011), 93. doi: 10.1007/978-3-0348-0069-3_6. [15] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, vol. 44 of Springer Series in Computational Mathematics,, Springer, (2013). [16] M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid,, J. Math. Fluid Mech., 9 (2007), 262. doi: 10.1007/s00021-005-0201-7. [17] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129. [18] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15 of Springer Series in Computational Mathematics,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1. [19] F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505. doi: 10.3934/dcdss.2011.4.505. [20] I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid,, Commun. Pure Appl. Anal., 13 (2014), 1759. doi: 10.3934/cpaa.2014.13.1759. [21] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. [22] P. Clément, Approximation by finite element functions using local regularization,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9 (1975), 77. [23] H. Cohen and S. I. Rubinow, Some mathematical topics in biology,, in Proc. Symp. on System Theory, (1965), 321. [24] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303. doi: 10.1007/s00205-005-0385-2. [25] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33. [26] M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations,, SIAM J. Math. Anal., 20 (1989), 74. doi: 10.1137/0520006. [27] G. de Vries and D. H. Norrie, The application of the finite-element technique to potential flow problems,, Journal of Applied Mechanics, 38 (1971), 798. doi: 10.1115/1.3408957. [28] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. [29] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Semidiscrete finite element approximations of a linear fluid-structure interaction problem,, SIAM J. Numer. Anal., 42 (2004), 1. doi: 10.1137/S0036142903408654. [30] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, vol. 159 of Applied Mathematical Sciences,, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4355-5. [31] M. Fortin, Calcul Numérique des écoulements des Fluides de Bingham et des Fluides Newtoniens Incompressibles par la Méthode des éléments Finis,, PhD thesis, (1972). [32] M. Fortin, An analysis of the convergence of mixed finite element methods,, RAIRO Anal. Numér., 11 (1977), 341. [33] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, 2nd edition, (2011). doi: 10.1007/978-0-387-09620-9. [34] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. [35] R. Glowinski and O. Pironneau, On a mixed finite element approximation of the Stokes problem. I. Convergence of the approximate solutions,, Numer. Math., 33 (1979), 397. doi: 10.1007/BF01399323. [36] C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid Mech., 4 (2002), 76. doi: 10.1007/s00021-002-8536-9. [37] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24,, Pitman, (1985). [38] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model,, Nonlinearity, 27 (2014), 467. doi: 10.1088/0951-7715/27/3/467. [39] S. Kesavan, Topics in Functional Analysis and Applications,, John Wiley & Sons Inc., (1989). [40] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction,, Indiana Univ. Math. J., 61 (2012), 1817. doi: 10.1512/iumj.2012.61.4746. [41] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969). [42] I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction,, Semigroup Forum, 82 (2011), 61. doi: 10.1007/s00233-010-9281-7. [43] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Springer-Verlag, (1972). [44] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod, (1969). [45] J. Liu, Open and traction boundary conditions for the incompressible Navier-Stokes equations,, J. Comput. Phys., 228 (2009), 7250. doi: 10.1016/j.jcp.2009.06.021. [46] H. Morand and R. Ohayon, Fluid-Structure Interaction: Applied Numerical Methods,, Wiley-Masson Series Research in Applied Mathematics, (1995). [47] B. Muha and S. Canić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls,, Arch. Ration. Mech. Anal., 207 (2013), 919. doi: 10.1007/s00205-012-0585-5. [48] B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition,, J. Differential Equations, 260 (2016), 8550. doi: 10.1016/j.jde.2016.02.029. [49] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle,, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 305. [50] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions,, Math. Comp., 54 (1990), 483. doi: 10.1090/S0025-5718-1990-1011446-7. [51] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid,, J. Math. Fluid Mech., 6 (2004), 53. doi: 10.1007/s00021-003-0083-4. [52] C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique,, Internat. J. Comput. & Fluids, 1 (1973), 73. doi: 10.1016/0045-7930(73)90027-3. [53] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations,, RAIRO Anal. Numér., 18 (1984), 175. [54] O. Zienkiewicz and Y. K. Cheung, Finite elements in the solution of field problems,, The Engineer, 220 (1965), 507.

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##### References:
 [1] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties,, in Fluids and waves, (2007), 15. doi: 10.1090/conm/440/08475. [2] G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system,, J. Differential Equations, 258 (2015), 4398. doi: 10.1016/j.jde.2015.01.037. [3] G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method,, Appl. Math. (Warsaw), 35 (2008), 259. doi: 10.4064/am35-3-2. [4] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417. doi: 10.3934/dcdss.2009.2.417. [5] O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, vol. 35 of Classics in Applied Mathematics,, Society for Industrial and Applied Mathematics (SIAM), (2001). doi: 10.1137/1.9780898719253. [6] I. Babuška, Error-bounds for finite element method,, Numer. Math., 16 (): 322. [7] I. Babuška and V. Nistor, Interior numerical approximation of boundary value problems with a distributional data,, Numer. Methods Partial Differential Equations, 22 (2006), 79. doi: 10.1002/num.20086. [8] I. Babuška, V. Nistor and N. Tarfulea, Approximate and low regularity Dirichlet boundary conditions in the generalized finite element method,, Math. Models Methods Appl. Sci., 17 (2007), 2115. doi: 10.1142/S0218202507002571. [9] V. Barbu, Z. Grujic, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55. doi: 10.1090/conm/440/08476. [10] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables,, Numer. Math., 33 (1979), 211. doi: 10.1007/BF01399555. [11] L. Bociu, S. Derochers and D. Toundykov, Linearized hydroelasticity: A numerical study,, To appear in: Evol. Equ. Control Theory, (2016). [12] L. Bociu, D. Toundykov and J.-P. Zolésio, Well-posedness analysis for the total linearization of a fluid-elasticity interaction,, SIAM J. Math. Anal., 47 (2015), 1958. doi: 10.1137/140970689. [13] L. Bociu and J.-P. Zolésio, Existence for the linearization of a steady state fluid/nonlinear elasticity interaction,, Discrete Contin. Dyn. Syst., 1 (2011), 184. [14] L. Bociu and J.-P. Zolésio, Linearization of a coupled system of nonlinear elasticity and viscous fluid,, in Modern aspects of the theory of partial differential equations, (2011), 93. doi: 10.1007/978-3-0348-0069-3_6. [15] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, vol. 44 of Springer Series in Computational Mathematics,, Springer, (2013). [16] M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid,, J. Math. Fluid Mech., 9 (2007), 262. doi: 10.1007/s00021-005-0201-7. [17] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129. [18] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, vol. 15 of Springer Series in Computational Mathematics,, Springer-Verlag, (1991). doi: 10.1007/978-1-4612-3172-1. [19] F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505. doi: 10.3934/dcdss.2011.4.505. [20] I. Chueshov, Interaction of an elastic plate with a linearized inviscid incompressible fluid,, Commun. Pure Appl. Anal., 13 (2014), 1759. doi: 10.3934/cpaa.2014.13.1759. [21] I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. [22] P. Clément, Approximation by finite element functions using local regularization,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9 (1975), 77. [23] H. Cohen and S. I. Rubinow, Some mathematical topics in biology,, in Proc. Symp. on System Theory, (1965), 321. [24] D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2006), 303. doi: 10.1007/s00205-005-0385-2. [25] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33. [26] M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations,, SIAM J. Math. Anal., 20 (1989), 74. doi: 10.1137/0520006. [27] G. de Vries and D. H. Norrie, The application of the finite-element technique to potential flow problems,, Journal of Applied Mechanics, 38 (1971), 798. doi: 10.1115/1.3408957. [28] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discrete Contin. Dyn. Syst., 9 (2003), 633. doi: 10.3934/dcds.2003.9.633. [29] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Semidiscrete finite element approximations of a linear fluid-structure interaction problem,, SIAM J. Numer. Anal., 42 (2004), 1. doi: 10.1137/S0036142903408654. [30] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, vol. 159 of Applied Mathematical Sciences,, Springer-Verlag, (2004). doi: 10.1007/978-1-4757-4355-5. [31] M. Fortin, Calcul Numérique des écoulements des Fluides de Bingham et des Fluides Newtoniens Incompressibles par la Méthode des éléments Finis,, PhD thesis, (1972). [32] M. Fortin, An analysis of the convergence of mixed finite element methods,, RAIRO Anal. Numér., 11 (1977), 341. [33] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations,, 2nd edition, (2011). doi: 10.1007/978-0-387-09620-9. [34] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5 of Springer Series in Computational Mathematics,, Springer-Verlag, (1986). doi: 10.1007/978-3-642-61623-5. [35] R. Glowinski and O. Pironneau, On a mixed finite element approximation of the Stokes problem. I. Convergence of the approximate solutions,, Numer. Math., 33 (1979), 397. doi: 10.1007/BF01399323. [36] C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem,, J. Math. Fluid Mech., 4 (2002), 76. doi: 10.1007/s00021-002-8536-9. [37] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24,, Pitman, (1985). [38] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model,, Nonlinearity, 27 (2014), 467. doi: 10.1088/0951-7715/27/3/467. [39] S. Kesavan, Topics in Functional Analysis and Applications,, John Wiley & Sons Inc., (1989). [40] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction,, Indiana Univ. Math. J., 61 (2012), 1817. doi: 10.1512/iumj.2012.61.4746. [41] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969). [42] I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes-elastic wave interaction,, Semigroup Forum, 82 (2011), 61. doi: 10.1007/s00233-010-9281-7. [43] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications,, Springer-Verlag, (1972). [44] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod, (1969). [45] J. Liu, Open and traction boundary conditions for the incompressible Navier-Stokes equations,, J. Comput. Phys., 228 (2009), 7250. doi: 10.1016/j.jcp.2009.06.021. [46] H. Morand and R. Ohayon, Fluid-Structure Interaction: Applied Numerical Methods,, Wiley-Masson Series Research in Applied Mathematics, (1995). [47] B. Muha and S. Canić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls,, Arch. Ration. Mech. Anal., 207 (2013), 919. doi: 10.1007/s00205-012-0585-5. [48] B. Muha and S. Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition,, J. Differential Equations, 260 (2016), 8550. doi: 10.1016/j.jde.2016.02.029. [49] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle,, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 305. [50] L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions,, Math. Comp., 54 (1990), 483. doi: 10.1090/S0025-5718-1990-1011446-7. [51] T. Takahashi and M. Tucsnak, Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid,, J. Math. Fluid Mech., 6 (2004), 53. doi: 10.1007/s00021-003-0083-4. [52] C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique,, Internat. J. Comput. & Fluids, 1 (1973), 73. doi: 10.1016/0045-7930(73)90027-3. [53] R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations,, RAIRO Anal. Numér., 18 (1984), 175. [54] O. Zienkiewicz and Y. K. Cheung, Finite elements in the solution of field problems,, The Engineer, 220 (1965), 507.
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