December  2014, 3(4): 557-578. doi: 10.3934/eect.2014.3.557

A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction

1. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, United States

2. 

Department of Mathematics, Statistics, and Computer Science, Dordt College, Sioux Center, IA 51250, United States

Received  February 2014 Revised  May 2014 Published  October 2014

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ coupled to a fourth order plate equation, possibly with rotational inertia parameter $\rho >0$. This plate PDE evolves on a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on $\Omega$ is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on $\Omega$. We note that as the Stokes fluid velocity does not vanish on $\Omega$, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational (``inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.
Citation: George Avalos, Thomas J. Clark. A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction. Evolution Equations and Control Theory, 2014, 3 (4) : 557-578. doi: 10.3934/eect.2014.3.557
References:
[1]

G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method, Applicationes Mathematicae, 35 (2008), 259-280. doi: 10.4064/am35-3-2.

[2]

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Theory and Computation, Academic Press, 1984.

[3]

H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Meth. in the Appl. Sci., 2 (1980), 556-581. doi: 10.1002/mma.1670020416.

[4]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994. doi: 10.1007/978-1-4757-4338-8.

[5]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-3172-1.

[6]

A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech, 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[7]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Communications on Pure and Applied Analysis, 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.

[8]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496.

[9]

P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.

[10]

M. Dauge, Stationary stokes and navier-stokes systems on two- or three-dimensional domains with corners, part 1: linearized equations, Siam J. Math. Anal., 20 (1989), 74-97. doi: 10.1137/0520006.

[11]

V. Domínguez and F. J. Sayas, Algorithm 884: A simple MATLAB implementation of the argyris element, ACM Trans. Math. Software, 35 (2009), Art. 16, 11 pp.

[12]

A. Ern and J. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-4355-5.

[13]

P. Grisvard, Caracterization de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[14]

B. Kellogg, Properties of solutions of elliptic boundary value problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations(ed. A. K. Aziz), Academic Press, New York, (1972), 47-81.

[15]

S. Kesavan, Topics in Functional Analysis and Applications, Wiley, New York, 1989.

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, 1972.

[17]

P. Šolin, Partial Differential Equations and the Finite Element Method, Wiley, 2006.

[18]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, Rhode Island, 2001.

[19]

R. Wait and A. R. Mitchell, Finite Element Analysis and Applications, Wiley, 1985.

show all references

References:
[1]

G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method, Applicationes Mathematicae, 35 (2008), 259-280. doi: 10.4064/am35-3-2.

[2]

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Theory and Computation, Academic Press, 1984.

[3]

H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Meth. in the Appl. Sci., 2 (1980), 556-581. doi: 10.1002/mma.1670020416.

[4]

S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994. doi: 10.1007/978-1-4757-4338-8.

[5]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-3172-1.

[6]

A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech, 7 (2005), 368-404. doi: 10.1007/s00021-004-0121-y.

[7]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Communications on Pure and Applied Analysis, 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.

[8]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812. doi: 10.1002/mma.1496.

[9]

P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.

[10]

M. Dauge, Stationary stokes and navier-stokes systems on two- or three-dimensional domains with corners, part 1: linearized equations, Siam J. Math. Anal., 20 (1989), 74-97. doi: 10.1137/0520006.

[11]

V. Domínguez and F. J. Sayas, Algorithm 884: A simple MATLAB implementation of the argyris element, ACM Trans. Math. Software, 35 (2009), Art. 16, 11 pp.

[12]

A. Ern and J. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, 2004. doi: 10.1007/978-1-4757-4355-5.

[13]

P. Grisvard, Caracterization de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[14]

B. Kellogg, Properties of solutions of elliptic boundary value problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations(ed. A. K. Aziz), Academic Press, New York, (1972), 47-81.

[15]

S. Kesavan, Topics in Functional Analysis and Applications, Wiley, New York, 1989.

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, 1972.

[17]

P. Šolin, Partial Differential Equations and the Finite Element Method, Wiley, 2006.

[18]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, Rhode Island, 2001.

[19]

R. Wait and A. R. Mitchell, Finite Element Analysis and Applications, Wiley, 1985.

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