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A priori error analysis of a fully-mixedfinite element method for a two-dimensional fluid-solid interaction problem∗

Published online by Cambridge University Press:  11 January 2013

Carolina Domínguez ,
Gabriel N. Gatica ,
Salim Meddahi  and
Ricardo Oyarzúa
Carolina Domínguez
Affiliation:
Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.. cdominguez@ing-mat.udec.cl
Gabriel N. Gatica
Affiliation:
CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.; ggatica@ing-mat.udec.cl
Salim Meddahi
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España.; salim@uniovi.es
Ricardo Oyarzúa
Affiliation:
CI2MA (Universidad de Concepción) and Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 3-C, Concepción, Chile.; royarzua@ubiobio.cl
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Abstract

We introduce and analyze a fully-mixed finite element method for a fluid-solidinteraction problem in 2D. The model consists of an elastic body which is subject to agiven incident wave that travels in the fluid surrounding it. Actually, the fluid issupposed to occupy an annular region, and hence a Robin boundary condition imitating thebehavior of the scattered field at infinity is imposed on its exterior boundary, which islocated far from the obstacle. The media are governed by the elastodynamic and acousticequations in time-harmonic regime, respectively, and the transmission conditions are givenby the equilibrium of forces and the equality of the corresponding normal displacements.We first apply dual-mixed approaches in both domains, and then employ the governingequations to eliminate the displacement u of the solid and the pressure pof the fluid. In addition, since both transmission conditions become essential, they areenforced weakly by means of two suitable Lagrange multipliers. As a consequence, theCauchy stress tensor and the rotation of the solid, together with the gradient ofp and the traces of u and p on the boundary of thefluid, constitute the unknowns of the coupled problem. Next, we show that suitabledecompositions of the spaces to which the stress and the gradient of pbelong, allow the application of the Babuška–Brezzi theory and the Fredholm alternativefor analyzing the solvability of the resulting continuous formulation. The unknowns of thesolid and the fluid are then approximated by a conforming Galerkin scheme defined in termsof PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, andcontinuous piecewise linear functions on the boundary. Then, the analysis of the discretemethod relies on a stable decomposition of the corresponding finite element spaces andalso on a classical result on projection methods for Fredholm operators of index zero.Finally, some numerical results illustrating the theory are presented.

Keywords

Mixed finite elementsHelmholtz equationelastodynamic equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 471 - 506
Copyright
© EDP Sciences, SMAI, 2013

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References

Références

Arnold, D.N., Brezzi, F. and Douglas, J., PEERS : A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. Google Scholar
I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method. in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A.K. Aziz. Academic Press, New York (1972).
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag New York, Inc. (1994).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991).
Bielak, J. and MacCamy, R.C., Symmetric finite element and boundary integral coupling methods for fluid-solid interaction. Quarterly Appl. Math. 49 (1991) 107–119. Google Scholar
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition. Springer-Verlag, Berlin (1998).
Gatica, G.N., Márquez, A. and Meddahi, S., Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid-solid interaction problem. SIAM J. Numer. Anal. 45 (2007) 2072–2097. Google Scholar
Gatica, G.N., Márquez, A. and Meddahi, S., A new dual-mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions. Comput. Methods Appl. Mech. Engrg. 197 (2008) 1115–1130. Google Scholar
Gatica, G.N., Márquez, A. and Meddahi, S., Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid-solid interaction problem. Appl. Numer. Math. 59 (2009) 2735–2750. Google Scholar
Gatica, G.N., Márquez, A. and Meddahi, S., 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. Google Scholar
G.N. Gatica, A. Márquez and S. Meddahi, Analysis of an augmented fully-mixed finite element method for a three-dimensional fluid-solid interaction problem. Preprint 2011-23, Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción (2011).
Gatica, G.N., Oyarzúa, R. and Sayas, F.J., Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comput. 80 276 (2011) 1911–1948. Google Scholar
Girault, V. and Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag. Springer Ser. Comput. Math. 5 (1986). Google Scholar
Grisvard, P., Elliptic Problems in Non-Smooth Domains. Pitman. Monogr. Studies Math. 24 (1985). Google Scholar
P. Grisvard, Problèmes aux limites dans les polygones. Mode d’emploi. EDF. Bulletin de la Direction des Etudes et Recherches (Serie C) 1 (1986) 21–59.
Hiptmair, R., Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. Google Scholar
G.C. Hsiao, On the boundary-field equation methods for fluid-structure interactions, edited by L. Jentsch and F. Tröltzsch, Teubner-Text zur Mathematik, Band, B.G. Teubner Veriagsgesellschaft, Stuttgart, in Probl. Methods Math. Phys. 34 (1994) 79–88. CrossRef
Hsiao, G.C., Kleinman, R.E. and Roach, G.F., Weak solutions of fluid-solid interaction problems. Math. Nachrichten 218 (2000) 139–163. Google Scholar
G.C. Hsiao, R.E. Kleinman and L.S. Schuetz, On variational formulations of boundary value problems for fluid-solid interactions, edited by M.F. McCarthy and M.A. Hayes. Elsevier Science Publishers B.V. (North-Holland), in Elastic Wave Propagation (1989) 321–326.
F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York (1998).
R. Kress, Linear Integral Equ. Springer-Verlag, Berlin (1989).
Lonsing, M. and Verfürth, R., On the stability of BDMS and PEERS elements. Numer. Math. 99 (2004) 131–140. Google Scholar
Márquez, A., Meddahi, S. and Selgas, V., A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems. J. Comput. Phys. 199 (2004) 205–220. Google Scholar
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000).
Meddahi, S. and Sayas, F.-J., Analysis of a new BEM-FEM coupling for two dimensional fluid-solid interaction. Numer. Methods Partial Differ. Equ. 21 (2005) 1017–1042. Google Scholar
J.E. Roberts and J.M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991).
Stenberg, R., A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513–538. Google Scholar