Thesis Summary:

The main purpose of this thesis is to develop the a priori and a posteriori error analyses of a fullymixed finite element method for a fluid-solid interaction problem in 2D. In addition, we also derive a reliable and efficient residual-based a posteriori error estimator for the plane linear elasticity problem with pure traction boundary conditions. First, we develop an a priori 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 timeharmonic regime, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We introduce dual-mixed approaches in both domains, which yields the stress and the rotation in the solid, as well as the pressure gradient in the fluid, as the main unknowns. In turn, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. Next, we show that suitable decompositions of the spaces to which the stress and the pressure gradient belong, allow the application of the Babuˇska-Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns are approximated by a conforming Galerkin scheme defined in terms of Raviart-Thomas element of lowest order in both domains, and continuous piecewise linear functions on the boundaries. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Next, as a preliminary analysis as well as a by product of this thesis, we consider the two-dimensional linear elasticity problem with non-homogeneous Neumann boundary conditions, and derive a reliable and efficient residual-based a posteriori error estimator for the corresponding stress-displacement-rotation dual-mixed variational formulation. The proof of reliability makes use of a suitable auxiliary problem, the continuous inf-sup conditions satisfied by the bilinear forms involved, and the local approximation properties of the Cl´ement and Raviart-Thomas interpolation operators. In turn, inverse and discrete trace inequalities, and the localization technique based on triangle-bubble and edge-bubble functions, are the main tools yielding the efficiency of the estimator. Finally, we derive a reliable and efficient residual-based a posteriori error estimator for the interaction problem studied in the first part. The main tools for proving the reliability of the estimator involve a global inf-sup condition, continuous and discrete Helmholtz decompositions on each domain, and the local approximation properties of the Cl´ement and Raviart-Thomas interpolation operators. Next, we apply the above mentioned techniques to obtain the efficiency. Finally, several numerical results confirming the reliability and efficiency of the estimator, and illustrating the good performance of the associated adaptive scheme, are reported.

ISI Publications from the Thesis

Carolina DOMINGUEZ, Gabriel N. GATICA, Antonio MARQUEZ: A residual-based a posteriori error estimator for the plane linear elasticity problem with pure traction boundary conditions. Journal of Computational and Applied Mathematics, vol. 292, pp. 486-504, (2016).

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).

Carolina DOMINGUEZ, Gabriel N. GATICA, Salim MEDDAHI, Ricardo OYARZúA: A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 47, 2, pp. 471-506, (2013).