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A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

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Abstract

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.

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Author information

Authors and Affiliations

  1. CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    George C. Hsiao

  3. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, Spain

    Salim Meddahi

Authors
  1. Gabriel N. Gatica
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  2. George C. Hsiao
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  3. Salim Meddahi
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Corresponding author

Correspondence to Gabriel N. Gatica.

Additional information

This research was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, and by Ministery of Education and Science of Spain through the Project MTM2007-65088.

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Gatica, G.N., Hsiao, G.C. & Meddahi, S. A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem. Numer. Math. 114, 63–106 (2009). https://doi.org/10.1007/s00211-009-0250-6

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  • DOI: https://doi.org/10.1007/s00211-009-0250-6

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