### A POSTERIORI ERROR ANALYSIS OF A FULLY-MIXED FINITE ELEMENT METHOD FOR A TWO-DIMENSIONAL FLUID-SOLID INTERACTION PROBLEM

Carolina Domínguez1, Gabriel N. Gatica2, Salim Meddahi3

1. 1 Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Avenida Eduardo Morales Miranda-Campus Isla Teja, Valdivia, Chile;
2 CI2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile;
3 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España
• Received:2014-02-18 Revised:2015-06-05 Online:2015-11-15 Published:2015-11-15
• Supported by:

This research was partially supported by BASAL project CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática(CI2 MA), Universidad de Concepción, and by CONICYT project Anillo ACT1118(ANANUM).

Carolina Domínguez, Gabriel N. Gatica, Salim Meddahi. A POSTERIORI ERROR ANALYSIS OF A FULLY-MIXED FINITE ELEMENT METHOD FOR A TWO-DIMENSIONAL FLUID-SOLID INTERACTION PROBLEM[J]. Journal of Computational Mathematics, 2015, 33(6): 606-641.

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. As the main contribution of this work, we derive a reliable and efficient residual-based a posteriori error estimator for the aforedescribed coupled problem. Some numerical results confirming the properties of the estimator are also reported.

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