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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.. [email protected]
Gabriel N. Gatica
Affiliation:
CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.; [email protected]
Salim Meddahi
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España.; [email protected]
Ricardo Oyarzúa
Affiliation:
CI2MA (Universidad de Concepción) and Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 3-C, Concepción, Chile.; [email protected]
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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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