## Preprint 2014-02

## Carolina Dominguez, Gabriel N. Gatica, Salim Meddahi:

### A posteriori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

### Abstract:

In this paper we develop an a posteriori error analysis of a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements, and the fluid is supposed to occupy an annular region surrounding the solid, so that a Robin boundary condition imitating the behavior of the Sommerfeld condition is imposed on its exterior boundary. Dual-mixed approaches are applied in both domains, and the governing equations are employed to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart-Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The main tools for proving the reliability of the estimator involve the continuous global inf-sup condition, continuous and discrete Helmholtz decompositions on each domain, and the local approximation properties of the Clement interpolant and Raviart-Thomas operator. Then, inverse inequalities, discrete trace 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.

This preprint gave rise to the following definitive publication(s):

**Carolina DOMINGUEZ, Gabriel N. GATICA, Salim MEDDAHI: ***A posteriori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem*. Journal of Computational Mathematics, vol. 33, 6, pp. 606-641, (2015).